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Equidistribution and counting under equilibrium statesin negatively curved spaces and graphs of groups.

Applications to non-Archimedean Diophantineapproximation

Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin

To cite this version:Anne Broise-Alamichel, Jouni Parkkonen, Frédéric Paulin. Equidistribution and counting under equi-librium states in negatively curved spaces and graphs of groups. Applications to non-ArchimedeanDiophantine approximation. 2016. hal-01421211

Equidistribution and counting under equilibrium states innegatively curved spaces and graphs of groups. Applications

to non-Archimedean Diophantine approximation

Anne Broise-Alamichel Jouni Parkkonen Frédéric Paulin

December 19, 2016

2 19/12/2016

Contents

1 Introduction 7

I Geometry and dynamics in negative curvature 23

2 Negatively curved geometry 252.1 General notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Background on CATp´1q spaces . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Generalised geodesic lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4 The unit tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.5 Normal bundles and dynamical neighbourhoods . . . . . . . . . . . . . . . . . 322.6 Creating common perpendiculars . . . . . . . . . . . . . . . . . . . . . . . . . 342.7 Metric and simplicial trees, and graphs of groups . . . . . . . . . . . . . . . . 35

3 Potentials, critical exponents and Gibbs cocycles 413.1 Background on (uniformly local) Hölder-continuity . . . . . . . . . . . . . . . 413.2 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Poincaré series and critical exponents . . . . . . . . . . . . . . . . . . . . . . . 453.4 Gibbs cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Systems of conductances on trees and generalised electrical networks . . . . . 50

4 Patterson-Sullivan and Bowen-Margulis measures with potential on CATp´1qspaces 534.1 Patterson densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Gibbs measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3 Patterson densities for simplicial trees . . . . . . . . . . . . . . . . . . . . . . 604.4 Gibbs measures for metric and simplicial trees . . . . . . . . . . . . . . . . . . 62

5 Symbolic dynamics of geodesic flows on trees 675.1 Two-sided topological Markov shifts . . . . . . . . . . . . . . . . . . . . . . . 675.2 Coding discrete time geodesic flows on simplicial trees . . . . . . . . . . . . . 685.3 Coding continuous time geodesic flows on metric trees . . . . . . . . . . . . . 785.4 The variational principle for metric and simplicial trees . . . . . . . . . . . . . 84

6 Random walks on weighted graphs of groups 916.1 Laplacian operators on weighted graphs of groups . . . . . . . . . . . . . . . . 91

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6.2 Patterson densities as harmonic measures for simplicialtrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7 Skinning measures with potential on CATp´1q spaces 1037.1 Skinning measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.2 Equivariant families of convex subsets and their skinning measures . . . . . . 108

8 Explicit measure computations for simplicial trees and graphs of groups 1118.1 Computations of Bowen-Margulis measures for simplicial trees . . . . . . . . . 1128.2 Computations of skinning measures for simplicial trees . . . . . . . . . . . . . 116

9 Rate of mixing for the geodesic flow 1219.1 Rate of mixing for Riemannian manifolds . . . . . . . . . . . . . . . . . . . . 1219.2 Rate of mixing for simplicial trees . . . . . . . . . . . . . . . . . . . . . . . . . 1229.3 Rate of mixing for metric trees . . . . . . . . . . . . . . . . . . . . . . . . . . 131

II Geometric equidistribution and counting 141

10 Equidistribution of equidistant level sets to Gibbs measures 14310.1 A general equidistribution result . . . . . . . . . . . . . . . . . . . . . . . . . 14310.2 Rate of equidistribution of equidistant level sets for manifolds . . . . . . . . . 14710.3 Equidistribution of equidistant level sets on simplicial

graphs and random walks on graphs of groups . . . . . . . . . . . . . . . . . . 14810.4 Rate of equidistribution for metric and simplicial trees . . . . . . . . . . . . . 152

11 Equidistribution of common perpendicular arcs 15911.1 Part I of the proof of Theorem 11.1: the common part . . . . . . . . . . . . . 16111.2 Part II of the proof of Theorem 11.1: the metric tree case . . . . . . . . . . . 16211.3 Part III of the proof of Theorem 11.1: the manifold case . . . . . . . . . . . . 16511.4 Equidistribution of common perpendiculars in simplicial trees . . . . . . . . . 172

12 Equidistribution and counting of common perpendiculars in quotient spaces18112.1 Multiplicities and counting functions in Riemannian orbifolds . . . . . . . . . 18112.2 Common perpendiculars in Riemannian orbifolds . . . . . . . . . . . . . . . . 18312.3 Error terms for equidistribution and counting for Riemannian orbifolds . . . . 18612.4 Equidistribution and counting for quotient simplicial and metric trees . . . . 19012.5 Counting for simplicial graphs of groups . . . . . . . . . . . . . . . . . . . . . 19612.6 Error terms for equidistribution and counting for metric and simplicial graphs

of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

13 Geometric applications 21113.1 Orbit counting in conjugacy classes for groups acting on trees . . . . . . . . . 21113.2 Equidistribution and counting of closed orbits on metric and simplicial graphs

(of groups) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

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III Arithmetic applications 219

14 Fields with discrete valuations 22114.1 Local fields and valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22114.2 Global function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

15 Bruhat-Tits trees and modular groups 22715.1 Bruhat-Tits trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22715.2 Modular graphs of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23015.3 Computations of measures for Bruhat-Tits trees . . . . . . . . . . . . . . . . . 23215.4 Exponential decay of correlation and error terms for arithmetic quotients of

Bruhat-Tits trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23615.5 Geometrically finite lattices with infinite Bowen-Margulis measure . . . . . . 244

16 Rational point equidistribution and counting in completed function fields 24716.1 Equidistribution of non-Archimedian Farey fractions . . . . . . . . . . . . . . 24716.2 Mertens’s formula in function fields . . . . . . . . . . . . . . . . . . . . . . . . 254

17 Equidistribution and counting of quadratic irrational points innon-Archimedean local fields 25717.1 Counting and equidistribution of loxodromic fixed points . . . . . . . . . . . . 25717.2 Counting and equidistribution of quadratic irrationals in positive characteristic 26117.3 Counting and equidistribution of quadratic irrationals in Qp . . . . . . . . . . 267

18 Counting and equidistribution of crossratios 27318.1 Counting and equidistribution of crossratios of loxodromic fixed points . . . . 27318.2 Counting and equidistribution of crossratios of quadratic irrationals . . . . . . 279

19 Counting and equidistribution of integral representations by quadratic normforms 281

Appendix 285

A A weak Gibbs measure is the unique equilibrium, by J. Buzzi 287A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287A.2 Proof of the main result Theorem A.4 . . . . . . . . . . . . . . . . . . . . . . 290

List of Symbols 297

Index 303

Bibliography 309

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Chapter 1

Introduction

In this book, we study equidistribution and counting problems concerning locally geodesic arcsin negatively curved spaces endowed with potentials, and we deduce, from the special case oftree quotients, various arithmetic applications to equidistribution and counting problems innon-Archimedean local fields.

For several decades, tools in ergodic theory and dynamical systems have been used to ob-tain geometric equidistribution and counting results on manifolds, both inspired by and withapplications to arithmetic and number theoretic problems, in particular in Diophantine ap-proximation. Especially pioneered by Margulis, this field has produced a huge corpus of works,by Bowen, Cosentino, Clozel, Dani, Einseidler, Eskin, Gorodnik, Ghosh, Guivarc’h, Kim,Kleinbock, Kontorovich, Lindenstraus, Margulis, McMullen, Michel, Mohammadi, Mozes,Nevo, Oh, Pollicott, Roblin, Shah, Sharp, Sullivan, Ullmo, Weiss and the last two authors, justto mention a few contributors. We refer for now to the surveys [Bab2, Oh, PaP16a, PaP16d]and we will explain in more details in this introduction the relation of our work with previousworks.

In this text, we consider geometric equidistribution and counting problems weighted witha potential function in quotient spaces of CATp´1q spaces by discrete groups of isometries.The CATp´1q spaces form a huge class of metric spaces that contains (but is not restricted to)metric trees, hyperbolic buildings and simply connected Riemannian manifolds with sectionalcurvature bounded above by ´1. See [BridH] and Chapter 2 for a review of some basicproperties of these spaces. Although some of the equidistribution and counting results withpotentials on negatively curved manifolds are known (see for instance [PauPS]), as well assome of such results on CATp´1q spaces without potential (see for instance [Rob2]), bringingtogether these two aspects and producing new results and applications is one of the goals ofthis book.

We extend the theory of Patterson-Sullivan, Bowen-Margulis and skinning measures toCATp´1q spaces with potentials, with a special emphasis on trees endowed with a system ofconductances. We prove that under natural nondegeneracy, mixing and finiteness assump-tions, the pushforward under the geodesic flow of the skinning measure of properly immersedlocally convex closed subsets of CATp´1q spaces equidistributes to the Gibbs measure, gen-eralising the main result of [PaP14a].

We also prove that the (appropriate generalisations of) the initial and terminal tangentvectors of the common perpendiculars to any two properly immersed locally convex closedsubsets jointly equidistribute to the skinning measures when the lengths of the common per-

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pendiculars tend to `8. This result is then used to prove asymptotic results on weightedcounting functions of common perpendiculars whose lengths tend to `8. Common perpendic-ulars have been studied, in various particular cases, sometimes not explicitly, by Basmajian,Bridgeman, Bridgeman-Kahn, Eskin-McMullen, Herrmann, Huber, Kontorovich-Oh, Mar-gulis, Martin-McKee-Wambach, Meyerhoff, Mirzakhani, Oh-Shah, Pollicott, Roblin, Shah,the last two authors and many others. See the comments after Theorem 1.5 below, and thesurvey [PaP16a] for references.

In the Part III of this book, we apply the geometric results obtained for trees to deducearithmetic applications in non-Archimedean local fields. In particular, we prove equidistri-bution and counting results for rationals and quadratic irrationals in any completion of anyfunction field over a finite field.

Let us now describe more precisely the content of this book, restricted to special cases forthe sake of the exposition.

Geometric and dynamical tools

Let Y be a geodesically complete connected proper locally CATp´1q space (or good orbispace),such that the fundamental group of Y is not virtually nilpotent. In this introduction, we willmainly concentrate on the cases where Y is either a metric graph (or graph of finite groupsin the sense of Bass and Serre, see [Ser3]) or a Riemannian manifold (or good orbifold) ofdimension at least 2 with sectional curvature at most ´1. Let GY be the space of locallygeodesic lines of Y , on which the geodesic flow pgtqtPR acts by real translations on the source.When Y is a simplicial1 graph (of finite groups), we consider the discrete time geodesic flowpgtqtPZ, see Section 2.7. If Y is a Riemannian manifold, then GY is naturally identified withthe unit tangent bundle T 1Y by the map that associates to a locally geodesic line its tangentvector at time 0. In general, we define T 1Y as the space of germs of locally geodesic lines inY , and GY maps onto T 1Y with possibly uncountable fibers.

Let F : T 1Y Ñ R be a continuous map, called a potential, which plays the same rolein the construction of Gibbs measures/equilibrium states as the energy function in Bowen’streatment of the thermodynamic formalism of symbolic dynamical systems in [Bowe2, Sect. 1].In this introduction, we assume that F is bounded in order to simplify the statements. Wedefine in Section 3.3 the critical exponent δF associated with F , which describes the loga-rithmic growth of an orbit of the fundamental group on the universal cover of Y weightedby the (lifted) potential F , and which coincides with the classical critical exponent whenF “ 0. When Y is a metric graph, we associate in Section 3.5 a potential Fc to a systemof conductances c (that is, a map from the set of edges of Y to R), in such a way that thecorrespondence c ÞÑ Fc is bijective at the level of cohomology classes, and we denote δFc byδc. We assume in the remainder of this introduction that δF is finite and positive.

We say that the pair pY, F q satisfies the HC-property if the integral of F on compact locallygeodesic segments of Y varies in a Hölder-continuous way on its extremities (see Definition3.4). The pairs which have the HC-property include Riemannian manifolds with pinchedsectional curvature at most ´1 and Hölder-continuous potentials, and metric graphs with anypotential. This HC-property is the new technical idea compared to [PauPS] which allows theextensions to our very general framework. See also [ConLT], under the strong assumptionthat Y is compact.

1that is, if its edges all have lengths 1

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In Chapter 4, building on the works of [Rob2] when F “ 0 and of [PauPS]2 when Y is aRiemannian manifold, we generalise, to locally CATp´1q spaces Y endowed with a potentialF satisfying the HC-property, the construction and basic properties of the Patterson densitiesat infinity of the universal cover of Y associated with F and the Gibbs measure mF on GYassociated with F .

Using the Patterson-Sullivan-Bowen-Margulis approach, the Patterson densities are limitsof renormalised measures on the orbit points of the fundamental group, weighted by thepotential, and the Gibbs measures on GY are local products of Patterson densities on theendpoints of the geodesic line, with the Lebesgue measure on the time parameter, weightedby the Gibbs cocycle defined by the potential.

Generalizing a result of [CoP2], we prove in Section 6.2 that when Y is a regular simpli-cial graph and c is an anti-reversible system of conductances, then the Patterson measures,normalized to be probability measures, are harmonic measures (or hitting measures) on B8Yfor a transient random walk on the vertices, whose transition probabilities are constructedusing the total mass of the Patterson measures.

Gibbs measures were first introduced in statistical mechanics, and are naturally associatedvia the thermodynamic formalism3 with symbolic dynamics. We prove in Section 4.2 that ourGibbs measures satisfy a Gibbs property analogous to the one in symbolic dynamics. If F “ 0,the Gibbs measure mF is the Bowen-Margulis measure mBM. If Y is a compact Riemannianmanifold and F is the strong unstable Jacobian v ÞÑ ´ d

dt |t“0ln Jac

`

gt|W´pvq

˘

pvq, then mF isthe Liouville measure and δF “ 0 (see [PauPS, Chap. 7] for more general assumptions on Y ).Thus, one interesting aspect of Gibbs measure is that they form a natural family of measuresinvariant under the geodesic flow that interpolates between the Liouville measure and theBowen-Margulis measure (which in variable curvature are in general not in the same measureclass). Another interesting point is that such measures are plentiful: a recent result of Belarif[Bel] proves that when Y is a geometrically finite Riemannian manifold with pinched nega-tive curvature and topologically mixing geodesic flow, the finite and mixing Gibbs measuresassociated with bounded Hölder-continuous potentials are, once normalised, dense (for theweak-star topology) in the whole space of probability measures invariant under the geodesicflow.

The Gibbs measures are remarkable measures for CATp´1q spaces endowed with poten-tials due to their unique ergodic-theoretic properties. Let pZ, pφtqtPRq be a topological spaceendowed with a continuous one-parameter group of homeomorphisms and let ψ : Z Ñ R be abounded continuous map. Let M be the set of Borel probability measures m on Z invariantunder the flow pφtqtPR. Let hmpφ1q be the (metric) entropy of the geodesic flow with respect tom P M . The metric pressure for ψ of a measure m P M and the pressure of ψ are respectively

Pψpmq “ hmpφ1q `

ż

Zψ dm and Pψ “ sup

mPMPψpmq .

An element m P M is an equilibrium state for ψ if the least upper bound defining Pψ isattained on m.

Let F 7 : GY Ñ R be the composition of the canonical map GY Ñ T 1Y with F , and notethat F 7 “ F if Y is a Riemannian manifold. When F “ 0 and Y is a Riemannian manifold,whose sectional curvatures and their first derivatives are bounded, by [OP, Thm. 2], the

2itself building on the works of Ledrappier [Led], Hamenstädt, Coudène, Mohsen3see for instance [Rue3, Kel, Zin]

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pressure PF coincides with the entropy of the geodesic flow, it is equal to the critical exponentof the fundamental group of Y , and the Bowen-Margulis measure mF “ mBM, normalisedto be a probability measure, is the measure of maximal entropy. When Y is a Riemannianmanifold whose sectional curvatures and their first derivatives are bounded and F is Hölder-continuous, by [PauPS, Thm. 6.1], we have PF “ δF . If furthermore the Gibbs measure mF

is finite and normalised to be a probability measure, then mF is an equilibrium state for F .We prove an analog of these results for the potential F 7 when Y is a metric graph of groups.

The case when Y is a finite simplicial graph4 is classical by the work of Bowen [Bowe2], asit reduces to arguments of subshifts of finite type (see for instance [CoP1]). When Y is acompact5 locally CATp´1q-space,6 a complete statement about existence, uniqueness andGibbs property of equilibrium states for any Hölder-continuous potential is given in [ConLT].

Theorem 1.1 (The variational principle for metric graphs of groups). Assume that Y is ametric graph of finite groups, with a positive lower bound and finite upper bound on the lengthsof edges. If the critical exponent δF is finite, if the Gibbs measure mF is finite, then PF 7 “ δFand the Gibbs measure normalised to be a probability measure is the unique equilibrium statefor F 7.

The main tool is a natural coding of the discrete time geodesic flow by a topological Markovshift (see Section 5.1). This coding is delicate when the vertex stabilisers are nontrivial, asin particular it does not satisfy in general the Markovian property of dependence only onthe immediate past (see Section 5.2). We then apply results of Buzzi and Sarig in symbolicdynamics over a countable alphabet (see Appendix A written by J. Buzzi), and suspensiontechniques introduced in Section 5.3.

In Chapter 7, we generalise for nonconstant potentials on any geodesically complete con-nected proper locally CATp´1q space Y the construction of the skinning measures σ`D andσ´D on the outer and inner unit normal bundles of a connected proper nonempty properlyimmersed closed locally convex subset D of Y . By definition, D is the image, by the universalcovering map, of a proper nonempty closed convex subset of the universal cover of Y , whosefamily of images under the universal covering group is locally finite. We refer to Section 2.5 forthe appropriate definition of the outer and inner unit normal bundles of D when the boundaryof D is not smooth. We construct these measures σ`D and σ´D as the induced measures onY of appropriate pushforwards of the Patterson densities associated with the potential F tothe outer and inner unit normal bundles of the lift of D in the universal cover of Y . Thisconstruction generalises the one in [PaP14a] when F “ 0, which itself generalises the one in[OhS1, OhS2] whenM has constant curvature and D is a ball, a horoball or a totally geodesicsubmanifold.

In Section 10.1, we prove the following result on the equidistribution of equidistant hyper-surfaces in CATp´1q spaces. This result is a generalisation of [PaP14a, Theo. 1] (valid in Rie-mannian manifolds with zero potential) which itself generalised the ones in [Mar2, EM, PaP12]when Y has constant curvature, F “ 0 and D is a ball, a horoball or a totally geodesic sub-manifold. See also [Rob2] when Y is a CATp´1q space, F “ 0 and D is a ball or a horoball.

4that is, a finite graph of trivial groups with edge lengths 15a very strong assumption that we do not want to make in this text6not in the orbifold sense, hence this excludes for instance the case of graphs of groups with some nontrivial

vertex stabiliser

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Theorem 1.2. Let pY, F q be a locally CATp´1q space endowed with a potential satisfying theHC-property. Assume that the Gibbs measure mF on GY is finite and mixing for the geodesicflow pgtqtPR, and that the skinning measure σ`D is finite and nonzero. Then, as t tends to`8, the pushforwards pgtq˚σ`D of the skinning measure of D by the geodesic flow weak-starconverges towards the Gibbs measure mF (after normalisation as probability measures).

We prove in Theorem 10.4 an analog of Theorem 1.2 for the discrete time geodesic flowon simplicial graphs and, more generally, simplicial graphs of groups. As a special case,we recover known results on nonbacktracking simple random walks on regular graphs. Theequidistribution of the pushforward of the skinning measure of a subgraph is a weighted versionof the following classical result, see for instance [ABLS], which under further assumptions onthe spectral properties on the graph gives precise rates of convergence.

Corollary 1.3. Let Y be a finite regular graph which is not bipartite. Let Y1 be a nonemptyconnected subgraph. Then the n-th vertex of the non-backtracking simple random walk on Ystarting transversally to Y1 converges in distribution to the uniform distribution as nÑ `8.

See Chapter 10 for more details and for the extensions to nonzero potential and to graphsof groups, as well as Section 10.4 for error terms.

The distribution of common perpendiculars

Let D´ and D` be connected proper nonempty properly immersed locally convex closedsubsets of Y . A common perpendicular from D´ to D` is a locally geodesic path in Y startingperpendicularly from D´ and arriving perpendicularly to D`.7 We denote the length of acommon perpendicular α from D´ to D` by λpαq, and its initial and terminal unit tangentvectors by v´α and v`α . In the general CATp´1q case, v˘α are two different parametrisations(by ¯r0, λpαqs) of α, considered as elements of the space

p

GY of generalised locally geodesiclines in Y , see [BartL] or Section 2.3. For all t ą 0, we denote by PerppD´, D`, tq the set ofcommon perpendiculars from D´ to D` with length at most t (considered with multiplicities),and we define the counting function with weights by

ND´, D`, F ptq “ÿ

αPPerppD´, D`, tq

eş

α F ,

whereş

α F “şλpαq0 F pgtv´α q dt. We refer to Section 12.1 for the definition of the multiplicities

in the manifold case, which are equal to 1 if D´ and D` are embedded and disjoint. Highermultiplicities for common perpendiculars α can occur for instance when D´ is a non-simpleclosed geodesic and the initial point of α is a multiple point of D´.

Let PerppD´, D`q be the set of all common perpendiculars from D´ to D` (consideredwith multiplicities). The family pλpαqqαPPerppD´, D`q is called the marked ortholength spec-trum from D´ to D`. The set of lengths (with multiplicities) of elements of PerppD´, D`qis called the ortholength spectrum of D´, D`. This second set has been introduced by Bas-majian [Basm] (under the name “full orthogonal spectrum”) when M has constant curvature,and D´ and D` are disjoint or equal embedded totally geodesic hypersurfaces or embeddedhorospherical cusp neighbourhoods or embedded balls. We refer to the paper [BridK] which

7See Section 2.6 for explanations when the boundary of D´ or D` is not smooth.

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proves that the ortholength spectrum with D˘ “ BM determines the volume of a compacthyperbolic manifold M with totally geodesic boundary (see also [Cal] and [MasM]).

We prove in Chapter 12 that the critical exponent δF of F is the exponential growth rateof ND´, D`, F ptq, and we give an asymptotic formula of the form ND´, D`, F ptq „ c eδF t astÑ `8, with error term estimates in appropriate situations. The constants c that will appearin such asymptotic formulas will be explicit, in terms of the measures naturally associatedwith the (normalised) potential F : the Gibbs measure mF and the skinning measures of D´

and D`.When F “ 0 and Y is a Riemannian manifold with pinched sectional curvature and finite

and mixing Bowen-Margulis measure, the asymptotics of the counting function ND´, D`, 0ptqare described in [PaP16c, Thm. 1]. The only restriction on the two convex sets D˘ is thattheir skinning measures are finite. Here, we generalise that result by allowing for nonzeropotential and more general CATp´1q spaces than just manifolds.

The counting function ND´, D`, 0ptq has been studied in negatively curved manifolds sincethe 1950’s and in a number of more recent works, sometimes in a different guise. A numberof special cases (all with F “ 0 and covered by the results of [PaP16c]) were known:• D´ and D` are reduced to points, by for instance [Hub2], [Mar1] and [Rob2],• D´ and D` are horoballs, by [BeHP], [HeP3], [Cos] and [Rob2] without an explicit form

of the constant in the asymptotic expression,• D´ is a point and D` is a totally geodesic submanifold, by [Her], [EM] and [OhS3] in

constant curvature,• D´ is a point and D` is a horoball, by [Kon] and [KonO] in constant curvature, and [Kim]

in rank one symmetric spaces,• D´ is a horoball and D` is a totally geodesic submanifold, by [OhS1] and [PaP12] in

constant curvature, and• D´ and D` are (properly immersed) locally geodesic lines in constant curvature and

dimension 3, by [Pol2].We refer to the survey [PaP16a] for more details on the manifold case.

When X is a compact metric or simplicial graph and D˘ are points, the asymptotics ofND´, D`, 0ptq as t Ñ `8 is treated in [Gui], as well as [Rob2]. Under the same setting, seealso the work of Kiro-Smilansky-Smilansky announced in [KiSS] for a counting result of paths(not assumed to be locally geodesic) in finite metric graphs with rationally independent edgelengths and vanishing potential.

The proofs of the asymptotic results on the counting function ND´, D`, F are based onthe following simultaneous equidistribution result that shows that the initial and terminaltangent vectors of the common perpendiculars equidistribute to the skinning measures of D´

and D`. We denote the unit Dirac mass at a point z by ∆z and the total mass of any measurem by m.

Theorem 1.4. Assume that Y is a nonelementary Riemannian manifold with pinched sec-tional curvature at most ´1 or a metric graph. Let F : T 1Y Ñ R be a potential, with finiteand positive critical exponent δF , which is bounded and Hölder-continuous when Y is a man-ifold. Let D˘ be as above. Assume that the Gibbs measure mF is finite and mixing for thegeodesic flow. For the weak-star convergence of measures on

p

GY ˆ

p

GY , we have

limtÑ`8

δF mF e´δF t

ÿ

αPPerppD´, D`, tq

eş

α F ∆v´αb∆v`α

“ σ`D´b σ´

D`.

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There is a similar statement for nonbipartite simplicial graphs and for more general graphsof groups on which the discrete time geodesic flow is mixing for the Gibbs measure, see theend of Chapter 11 and Section 12.4. Again, the results can then be interpreted in terms ofnonbacktracking random walk.

In Section 12.2, we deduce our counting results for common perpendiculars of the subsetsD´ and D` from the above simultaneous equidistribution theorem.

Theorem 1.5. (1) Let Y, F,D˘ be as in Theorem 1.4. Assume that the Gibbs measure mF

is finite and mixing for the continuous time geodesic flow and that the skinning measures σ`D´

and σ´D`

are finite and nonzero. Then, as sÑ `8,

ND´, D`, F psq „σ`D´ σ´

D`

mF

eδF s

δF.

(2) If Y is a finite nonbipartite simplicial graph, then

ND´, D`,F pnq „eδF σ`

D´ σ´

D`

peδF ´ 1q mF eδF n .

The above Assertion (1) is valid when Y is a good orbifold instead of a manifold ora metric graph of finite groups instead of a metric graph (for the appropriate notion ofmultiplicities), and when D´ and D` are replaced by locally finite families. See Section 12.4for generalisations of Assertion (2) to (possibly infinite) simplicial graphs of finite groups andSections 12.3 and 12.6 for error terms.

We avoid any compactness assumption on Y , we only assume that the Gibbs measure mF

of F is finite and that it is mixing for the geodesic flow. By Babillot’s theorem [Bab1], ifthe length spectrum of Y is not contained in a discrete subgroup of R, then mF is mixing iffinite. If Y is a Riemannian manifold, this condition is satisfied for instance if the limit setof a fundamental group of Y is not totally disconnected, see for instance [Dal1, Dal2]. WhenY is a metric graph, Babillot’s mixing condition is in particular satisfied if the lengths of theedges of Y are rationally independent.

As in [PaP16c], we have very weak finiteness and curvature assumptions on the spaceand the convex subsets we consider. Furthermore, the space Y is no longer required to be amanifold and we extend the theory to non-constant weights using equilibrium states. Suchweighted counting has only been used in the orbit-counting problem in manifolds with pinchednegative curvature in [PauPS]. The approach is based on ideas from Margulis’s thesis to usethe mixing of the geodesic flow. Our measures are much more general. As in [PaP16c], wepush simultaneously the unit normal vectors to the two convex sets D´ and D` in oppositedirections.

Classically, an important characterization of the Bowen-Margulis measure on closed neg-atively curved Riemannian manifolds (F “ 0) is that it coincides with the weak-star limitof properly normalised sums of Lebesgue measures supported on periodic orbits. The resultwas extended to CATp´1q spaces with zero potential in [Rob2] and to Gibbs measures inthe manifold case in [PauPS, Thm. 9.11]. As a corollary of the simultaneous equidistributionresult Theorem 1.4, we obtain a weighted version for simplicial and metric graphs of groups.The following is a simplified version of such a result for Gibbs measures of metric graphs.

Let Per1ptq be the set of prime periodic orbits of the geodesic flow on Y . Let λpgq denotethe length of a closed orbit g. Let Lg be the Lebesgue measure along g and let LgpF

7q bethe period of g for the potential F .

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Theorem 1.6. Assume that Y is a finite metric graph, that the critical exponent δF is positiveand that the Gibbs measure mF is mixing for the (continuous time) geodesic flow. As tÑ `8,the measures

δF eδF t

ÿ

gPPer1ptq

eLgpF qLg

andδF t e

δF tÿ

gPPer1ptq

eLgpF q Lg

λpgq

converge to mFmF

for the weak-star convergence of measures.

See Section 13.2 for the proof of the full result and for a similar statement for (possiblyinfinite) simplicial graphs of finite groups. As a corollary, we obtain counting results of simpleloops in metric and simplicial graphs, generalising results of [ParP], [Gui].

Corollary 1.7. Assume that Y is a finite metric graph with all vertices of degree at least 3such that the critical exponent δF is positive.

(1) If the Gibbs measure is mixing for the geodesic flow, then

ÿ

gPPer1ptq

eLgpF q „eδF t

δF t

as tÑ `8.

(2) If Y is simplicial and if the Gibbs measure is mixing for the discrete time geodesic flow,then

ÿ

gPPer1ptq

eLgpF q „eδF

eδF ´ 1

eδF t

t

as tÑ `8.

In the cases when error bounds are known for the mixing property of the continuous time ordiscrete time geodesic flow on GY , we obtain corresponding error terms in the equidistributionresult of Theorem 1.2 generalising [PaP14a, Theo. 20] (where F “ 0) and in the approximationof the counting function ND´, D`, 0 by the expression introduced in Theorem 1.5. In themanifold case, see [KM1], [Clo], [Dol1], [Sto], [Live], [GLP], and Section 12.3 for definitionsand precise references. Here is an example of such a result in the manifold case.

Theorem 1.8. Assume that Y is a compact Riemannian manifold and mF is exponentiallymixing under the geodesic flow for the Hölder regularity, or that Y is a locally symmetricspace, the boundary of D˘ is smooth, mF is finite, smooth, and exponentially mixing underthe geodesic flow for the Sobolev regularity. Assume that the strong stable/unstable ball massesby the conditionals of mF are Hölder-continuous in their radius.

(1) As t tends to `8, the pushforwards pgtq˚σ`D´ of the skinning measure of D´ by thegeodesic flow equidistribute towards the Gibbs measure mF with exponential speed.

(2) There exists κ ą 0 such that, as tÑ `8,

ND´, D`, F ptq “σ`D´ σ´

D`

δF mF eδF t

`

1`Ope´κtq˘

.

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See Section 12.3 for a discussion of the assumptions and the dependence of Op¨q on thedata. Similar (sometimes more precise) error estimates were known earlier for the countingfunction in special cases of D˘ in constant curvature geometrically finite manifolds (often insmall dimension) through the work of Huber, Selberg, Patterson, Lax and Phillips [LaxP],Cosentino [Cos], Kontorovich and Oh [KonO], Lee and Oh [LeO].

When Y is a finite volume hyperbolic manifold and the potential F is constant 0, the Gibbsmeasure is proportional to the Liouville measure and the skinning measures of totally geodesicsubmanifolds, balls and horoballs are proportional to the induced Riemannian measures ofthe unit normal bundles of their boundaries. In this situation, there are very explicit formsof the counting results in finite-volume hyperbolic manifolds, see [PaP16c, Cor.21], [PaP16a].These results are extended to complex hyperbolic space in [PaP16b].

As an example of this result, if D´ and D` are closed geodesics of Y of lengths `´ and ``,respectively, then the number N psq of common perpendiculars (counted with multiplicity)from D´ to D` of length at most s satisfies, as sÑ `8,

ND´, D`, 0psq „πn2´1Γpn´1

2 q2

2n´2pn´ 1qΓpn2 q

`´``VolpY q

epn´1qs . (1.1)

Counting in weighted graphs of groups

From now on in this introduction, we only consider metric or simplicial graphs or graphs ofgroups.

Let Y be a connected finite graph with set of vertices V Y and set of edges EY (see [Ser3]for the conventions). We assume that the degree of the graph at each vertex is at least 3. Letλ : EY Ñ s0,`8r with λpeq “ λpeq for every e P EYq be an edge length map, let Y “ |Y|λbe the geometric realisation of Y where the geometric realisation of every edge e P EY haslength λpeq, and let c : EY Ñ R be a map, called a (logarithmic) system of conductances inthe analogy between graphs and electrical networks, see for instance [Zem].

Let Y˘ be proper nonempty subgraphs of Y. For every t ě 0, we denote by PerppY´,Y`, tqthe set of edge paths α “ pe1, . . . , ekq in Y without backtracking, of length λpαq “

řki“1 λpeiq

at most t, of conductance cpαq “řki“1 cpeiq, starting from a vertex of Y´ but not by an edge

of Y´, ending at a vertex of Y` but not by an edge of Y`. Let

NY´,Y`ptq “ÿ

αPPerppY´,Y`, tq

ecpαq

be the number of paths without backtracking from Y´ to Y` of length at most t, countedwith weights defined by the system of conductances.

Recall that a real number x is Diophantine if it is badly approximable by rational numbers,that is, if there exist α, β ą 0 such that |x´ p

q | ě α q´β for all p, q P Z with q ą 0. We obtainthe following result, which is a very simplified version of our results for the sake of thisintroduction.

Theorem 1.9. (1) If Y has two cycles whose ratio of lengths is Diophantine, then there existsC ą 0 such that for every k P N´ t0u, as tÑ `8,

NY´,Y`ptq “ C eδc t`

1`Opt´kq˘

.

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(2) If λ ” 1, then there exists C 1, κ ą 0 such that, as n P N tends to `8,

NY´,Y`pnq “ C 1 eδc n`

1`Ope´κnq˘

.

Note that the Diophantine assumption on Y in Theorem 1.9 (1) is standard in the theoryof quantum graphs (see for instance [BerK]).

The constants C “ CY˘, c, λ ą 0 and C 1 “ C 1Y˘, c ą 0 in the above asymptotic formulasare explicit. When c ” 0 and λ ” 1, the constants can often be determined concretely,as indicated in the two examples below.8 Among the ingredients in these computations arethe explicit expressions of the total mass of many Bowen-Margulis measures and skinningmeasures obtained in Chapter 8.

See Sections 12.4, 12.5 and 12.6 for generalisations of Theorem 1.9 when the graphs Y˘are not embedded in Y, and for versions in (possibly infinite) metric graphs of finite groups.In particular, Assertion (2) remains valid if Y is the quotient of a uniform simplicial tree bya geometrically finite lattice in the sense of [Pau2], such as an arithmetic lattice in PGL2

over a non-Archimedian local field, see [Lub1]. Recall that a locally finite metric tree X isuniform if it admits a discrete and cocompact group of isometries, and that a lattice Γ of Xis a lattice in the locally compact group of isometries of X preserving without inversions thesimplicial structure. We refer for instance to [BasK, BasL] for uncountably many examplesof tree lattices.

Example 1.10. (1) When Y is a pq ` 1q-regular finite graph with constant edge length mapλ ” 1 and vanishing system of conductances c ” 0, then δc “ ln q, and if furthermore Y` andY´ are vertices, then (see Equation (12.10))

C 1 “q ` 1

pq ´ 1qCardpV Yq.

(2) When Y is biregular of degrees p` 1 and q` 1 with p, q ě 2, when λ ” 1 and c ” 0, thenδc “ ln

?pq , and if furthermore the subgraphs Y˘ are simple cycles of lengths L˘, then (see

Equation (12.11))

C 1 “p?q `

?pq2 L´ L`

2 ppq ´ 1q CardpEYq.

The main tool in order to obtain the error terms in Theorem 1.9 and its more generalversions is to study the error terms in the mixing property of the geodesic flow. Using thealready mentioned coding (given in Section 5.2) of the discrete time geodesic flow by a two-sided topological Markov shift, classical reduction to one-sided topological Markov shift, andresults of Young [You1] on the decay of correlations for Young towers with exponentiallysmall tails, we in particular obtain the following simple criteria for the exponential decay ofcorrelation of the discrete time geodesic flow. See Theorem 9.1 for the complete result. Inparticular, we do not assume Y to be finite.

Theorem 1.11. Assume that the Gibbs measure mF is finite and mixing for the discrete timegeodesic flow on Y. Assume moreover that there exist a finite subset E of V Y and C 1, κ1 ą 0such that for all n P N, we have

mF

`

t` P GY : `p0q P E and @ k P t1, . . . , nu, `pkq R Eu˘

ď C 1 e´κ1n .

Then the discrete time geodesic flow has exponential decay of Hölder correlations for mF .8See Section 12.4 for more examples.

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The assumption of having exponentially small mass of geodesic lines which have a bigreturn time to a given finite subset of V Y is in particular satisfied (see Section 9.2) if Y is thequotient of a uniform simplicial tree by a geometrically finite lattice in the sense of [Pau2],such as an arithmetic lattice in PGL2 over a non-Archimedian local field, see [Lub1], but alsoby many other examples of Y.

These results allow to prove in Section 9.3, under Diophantine assumptions, the rapidmixing property for the continuous time geodesic flow, that leads to Assertion (1) of Theorem1.9, see Section 12.6. The proof uses suspension techniques due to Dolgopyat [Dol2] when Yis a compact metric tree, and to Melbourne [Mel1] otherwise.

As a corollary of the general version of the counting result Theorem 1.5, we have thefollowing asymptotic for the orbital counting function in conjugacy classes for groups actingon trees. Given x0 P X and a nontrivial conjugacy class K in a discrete group Γ of isometriesof X, we consider the counting function

NK, x0ptq “ Cardtγ P K : dpx0, γx0q ď tu ,

introduced by Huber [Hub1] when X is replaced by the real hyperbolic plane and Γ is alattice. We refer to [PaP15] for many results on the asymptotic growth of such orbital countingfunctions in conjugacy classes, when X is replaced by a finitely generated group with a wordmetric, or a complete simply connected pinched negatively curved Riemannian manifold. Seealso [ChaP, ArCT].

Theorem 1.12. Let X be a uniform metric tree with vertices of degree ě 3, let δ be theHausdorff dimension of B8X, let Γ be a discrete group of isometries of X, let x0 be a vertexof X with trivial stabiliser in Γ, and let K be a loxodromic conjugacy class in Γ.(1) If the metric graph ΓzX is compact and has two cycles whose ratio of lengths is Diophan-tine, then there exists C ą 0 such that for every k P N´ t0u, as tÑ `8,

NK, x0ptq “ C eδ2t`

1`Opt´kq˘

.

(2) If X is simplicial and Γ is a geometrically finite lattice of X, then there exist C 1, κ ą 0such that, as n P N tends to `8,

NK, x0pnq “ C 1 eδ tpn´λpγqq2u`

1`Ope´κnq˘

.

We refer to Theorem 13.1 for a more general version, including a version with a systemof conductances in the counting function, and when K is elliptic. When ΓzX is compactand Γ is torsion free,9 Assertion (1) of this result is due to Kenison and Sharp [KeS], whoproved it using transfer operator techniques for suspensions of subshifts of finite type. Upto strengthening the Diophantine assumption, using work of Melbourne [Mel1] on the decayof correlations of suspensions of Young towers, we are able to extend Assertion (1) to allgeometrically finite lattices Γ of X in Chapter 13.1.

The constants C “ CK,x0 and C 1 “ C 1K,x0are explicit. For instance in Assertion (2), if X

is the geometric realisation of a regular simplicial tree X of degree q ` 1, if x0 is a vertex ofX, if K is the conjugacy class of γ0 with translation length λpγ0q on X, if

VolpΓzzXq “ÿ

rxsPΓzV X

1

|Γx|

9in particular Γ then has the very restricted structure of a free group

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is the volume10 of the quotient graph of groups ΓzzX , then

C 1 “λpγ0q

rZΓpγ0q : γZ0 s VolpΓzzXq,

where ZΓpγ0q is the centraliser of γ0 in Γ. When furthermore Γ is torsion free and ΓzX isfinite, as δ “ ln q, we get

NK, x0pnq “λpγ0q

CardpΓzXqqtpn´λpγ0qq2u `Opqp1´κ

1qn2q

as n P N tends to `8, thus recovering the result of [Dou] who used the spectral theory of thediscrete Laplacian.

Selected arithmetic applications

We end this introduction by giving a sample of our arithmetic applications (see Part III of thisbook) of the ergodic and dynamical results on the discrete time geodesic flow on simplicialtrees described in Part II of this book, as summarized above. Our equidistribution andcounting results of common perpendiculars between subtrees indeed produce equidistributionand counting results of rationals and quadratic irrationals in non-Archimedean local fields.We refer to [BrPP] for an announcement of the results of Part III, with a presentation differentfrom the one in this introduction.

To motivate what follows, consider R “ Z the ring of integers, K “ Q its field of fractions,pK “ R the completion of Q for the usual Archimedean absolute value | ¨ |, and Haar

pKthe

Lebesgue measure of R (which is the Haar measure of the additive group R normalised sothat Haar

pKpr0, 1sq “ 1).

The following equidistribution result of rationals, due to Neville [Nev], is a quantitativestatement on the density of K in pK: For the weak-star convergence of measures on pK, assÑ `8, we have

limsÑ`8

π

6s´2

ÿ

p,qPR : pR`qR“R, |q|ďs

∆ pq“ Haar

pK.

Furthermore, there exists ` P N such that for every smooth function ψ : pK Ñ C withcompact support, there is an error term in the above equidistribution claim evaluated onψ, of the form Opspln sqψ`q where ψ` is the Sobolev norm of ψ. The following countingresult due to Mertens on the asymptotic behaviour of the average of Euler’s totient functionϕ : k ÞÑ CardpRkRqˆ, follows from the above equidistribution one:

nÿ

k“1

ϕpkq “3

πn2 `Opn lnnq .

See [PaP14b] for an approach using methods similar to the ones in this text, and for instance[HaW, Th. 330] for a more traditional proof, as well as [Wal] for a better error term.

Let us now switch to a non-Archimedean setting, restricting to positive characteristic inthis introduction. See Part III for analogous applications in characteristic zero.

10See for instance [BasK, BasL].

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Let Fq be a finite field of order q. Let R “ FqrY s be the ring of polynomials in onevariable Y with coefficients in Fq. Let K “ FqpY q be the field of rational fractions in Y withcoefficients in Fq, which is the field of fractions of R. Let pK “ FqppY ´1qq be the field of formalLaurent series in the variable Y ´1 with coefficients in Fq, which is the completion of K forthe (ultrametric) absolute value |PQ | “ qdegP´degQ. Let O “ FqrrY ´1ss be the ring of formalpower series in Y ´1 with coefficients in Fq, which is the ball of centre 0 and radius 1 in pK forthis absolute value.

Note that pK is locally compact, and we endow the additive group pK with the Haar measureHaar

pKnormalised so that Haar

pKpOq “ 1. The following results extend (with appropriate

constants) when K is replaced by any function field of a nonsingular projective curve over Fqand pK any completion of K, see Part III.

The following equidistribution result11 of elements of K in pK gives an analog of Neville’sequidistribution results for function fields. Note that when G “ GL2pRq, we have pP,Qq PGp1, 0q if and only if xP,Qy “ R. We denote by Hx the stabiliser of any element x of any setendowed with any action of any group H.

Theorem 1.13. Let G be any finite index subgroup of GL2pRq. For the weak-star convergenceof measures on pK, we have

limtÑ`8

pq ` 1q rGL2pRq : Gs

pq ´ 1q q2 rGL2pRqp1,0q : Gp1,0qsq´2 t

ÿ

pP,QqPGp1,0q, degQďt

∆PQ“ Haar

pK.

We emphazise the fact that we are not assumingG to be a congruence subgroup of GL2pRq.This is made possible by our geometric and ergodic methods.

The following variation of this result is more interesting when the class number of thefunction field K is larger than 1 (see Corollary 16.7 in Chapter 16).

Theorem 1.14. Let m be a nonzero fractional ideal of R with norm Npmq. For the weak-starconvergence of measures on pK, we have

limtÑ`8

q ` 1

pq ´ 1q q2s´2

ÿ

px,yqPmˆmNpmq´1 Npyqďs, Rx`Ry“m

∆xy“ Haar

pK.

In the next two statements, we assume that the characteristic of K is different from 2. Ifα P pK is quadratic irrational overK,12 let ασ be the Galois conjugate of α,13 let trpαq “ α`ασ

and npαq “ αασ, and let

hpαq “1

|α´ ασ|.

This is an appropriate complexity for quadratic irrationals in a given orbit by homographiesunder PGL2pRq. See Section 17.2 and for instance [HeP4, §6] for motivations and results.Note that although there are only finitely many orbits by homographies of PGL2pRq on K(and exactly one in the particular case of this introduction), there are infinitely many orbitsof PGL2pRq in the set of quadratic irrationals in pK over K. The following result gives inparticular that any orbit of quadratic irrationals under PGL2pRq equidistributes in pK, whenthe complexity tends to infinity. See Theorem 17.5 in Section 17.2 for a more general version.We denote by ¨ the action by homographies of GL2p pKq on P1p pKq “ pK Y t8 “ r1 : 0su.

11See Theorem 16.4 in Chapter 16 for a more general version.12that is, α does not belong to K and satisfies a quadratic equation with coefficients in K13that is, the other root of the polynomial defining α

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Theorem 1.15. Let G be a finite index subgroup of GL2pRq. Let α0 P pK be a quadraticirrational over K. For the weak-star convergence of measures on pK, we have

limsÑ`8

pln qq pq ` 1q m0 rGL2pRq : Gs

2 q2 pq ´ 1q3ˇ

ˇ ln | tr g0|ˇ

ˇ

s´1ÿ

αPG¨α0, hpαqďs

∆α “ HaarpK.

where g0 P G fixes α0 with | tr g0| ą 1, and m0 is the index of gZ0 in Gα0.

Another equidistribution result of an orbit of quadratic irrationals under PGL2pRq isobtained by taking another complexity, constructed using crossratios with a fixed quadraticirrational. We denote by ra, b, c, ds “ pc´aqpd´bq

pc´bqpd´aq the crossratio of four pairwise distinct elements

in pK. If α, β P pK are two quadratic irrationals over K such that α R tβ, βσu,14 let

hβpαq “ maxt|rα, β, βσ, ασs|, |rασ, β, βσ, αs|u ,

which is also an appropriate complexity when α varies in a given orbit of quadratic irra-tionals by homographies under PGL2pRq. See Section 18.1 and for instance [PaP14b, §4] formotivations and results in the Archimedean case.

Theorem 1.16. Let G be a finite index subgroup of GL2pRq. Let α0, β P pK be two quadraticirrationals over K. For the weak-star convergence of measures on pK ´tβ, βσu, we have, withg0 and m0 as in the statement of Theorem 1.15,

limsÑ`8

pln qq pq ` 1q m0 rGL2pRq : Gs

2 q2 pq ´ 1q3 |β ´ βσ|ˇ

ˇ ln | tr g0|ˇ

ˇ

s´1ÿ

αPG¨α0, hβpαqďs

∆α “dHaar

pKpzq

|z ´ β| |z ´ βσ|.

The fact that the measure towards which we have an equidistribution is only absolutelycontinuous with respect to the Haar measure is explained by the invariance of α ÞÑ hβpαqunder the (infinite) stabiliser of β in PGL2pRq. See Theorem 18.4 in Section 18.1 for a moregeneral version.

The last statement of this introduction is an equidistribution result for the integral repre-sentations of quadratic norm forms

px, yq ÞÑ npx´ yαq

on K ˆ K, where α P pK is a quadratic irrational over K. See Theorem 19.1 in Section 19for a more general version, and for instance [PaP14b, §5.3] for motivations and results in theArchimedean case.

There is an extensive bibliography on the integral representation of norm forms and moregenerally decomposable forms over function fields, we only refer to [Sch1, Maso1, Gyo, Maso2].These references mostly consider higher degrees, with an algebraically closed ground field ofcharacteristic 0, instead of Fq.

Theorem 1.17. Let G be a finite index subgroup of GL2pRq and let β P Kv be a quadraticirrational over K. For the weak-star convergence of measures on pK ´ tβ, βσu, we have

limsÑ`8

pq ` 1q rGL2pRvq : Gs

q2 pq ´ 1q3 rGL2pRvqp1,0q : Gp1,0qss´1

ÿ

px,yqPGp1,0q,|x2´xy trpβq`y2 npβq|ďs

∆xy“

dHaarpKpzq

|z ´ β| |z ´ βσ|.

14See Section 18.1 when this condition is not satisfied.

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Furthermore, we have error estimates in the arithmetic applications: There exists κ ą 0such that for every locally constant function with compact support ψ : pK Ñ C in Theorems1.13, 1.14 and 1.15, or ψ : pK ´ tβ, βσu Ñ C in Theorems 1.16 and 1.17, there are error termsin the above equidistribution claims evaluated on ψ, of the form Ops´κq where s “ qt inTheorem 1.13, with for each result an explicit control on the test function ψ involving onlysome norm of ψ, see in particular Section 15.4.

The link between the geometry described in the first part of this introduction and theabove arithmetic statements is provided by the Bruhat-Tits tree of pPGL2, pKq, see [Ser3] andSection 15.1 for background. We refer to Part III for more general arithmetic applications.

Acknowledgements: This work was partially supported by the NSF grants no 093207800 and DMS-1440140, while the third author was in residence at the MSRI, Berkeley CA, during the Spring 2015and Fall 2016 semesters. The second author thanks Université Paris-Sud, Forschungsinstitut fürMathematik of ETH Zürich, and Vilho, Yrjö ja Kalle Väisälän rahasto for their support during thepreparation of this work. This research was supported by the CNRS PICS n0 6950 “Equidistributionet comptage en courbure négative et applications arithmétiques”. We thank, for interesting discussionson this text, Y. Benoist, J. Buzzi (for his help in Sections 5.2 and 9.2, and for kindly agreeing to insertAppendix A used in Section 5.4), N. Curien, S. Mozes, M. Pollicott (for his help in Section 9.3),R. Sharp and J.-B. Bost (for his help in Section 14.2). We especially thank O. Sarig for his help inSection 9.2: In a long email, he explained to us how to prove Theorem 9.2.

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Part I

Geometry and dynamics in negativecurvature

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Chapter 2

Negatively curved geometry

2.1 General notation

Here is some general notation that will be used in this text. We recommend the use of theList of symbols (mostly in alphabetical order by the first letter) and of the Index for easyreferences to the definitions in the text.

Let A be a subset of a set E. We denote by 1A : E Ñ t0, 1u the characteristic (orindicator) function of A: 1Apxq “ 1 if x P A, and 1Apxq “ 0 otherwise. We denote bycA “ E ´A the complementary subset of A in E.

We denote by ln the natural logarithm (with lnpeq “ 1).We denote by µ the total mass of a finite positive measure µ.If pX,A q and pY,Bq are measurable spaces, f : X Ñ Y a measurable map, and µ a

measure on X, we denote by f˚µ the image measure of µ by f , with f˚µpBq “ µpf´1pBqq forevery B P B.

If pX, dq is a metric space, then Bpx, rq is the closed ball with centre x P X and radiusr ą 0.

For every subset A of a metric space and for every ε ą 0, we denote by NεA the closedε-neighbourhood of A, and by convention N0A “ A.

If X is a uniquely geodesic space and x, y P X, then rx, ys is the unique geodesic segmentfrom x to y.

Given a topological space Z, we denote by CcpZq the vector space of continuous mapsfrom Z to R with compact support.

The negative part of a real-valued map f is f´ “ maxt0,´fu.We denote by ∆x the unit Dirac mass at a point x in any measurable space.

Finally, the symbol l right at the end of a statement indicates that this statement won’tbe given a proof, either since a reference is given or since it is an immediate consequence ofprevious statements.

2.2 Background on CATp´1q spaces

LetX be a geodesically complete proper CATp´1q space, let x0 P X be an arbitrary basepoint,and let Γ be a nonelementary discrete group of isometries of X.

We refer for example to [BridH] for the relevant definitions and complements on thesenotions. Recall that a geodesic metric space X is geodesically complete (or has extendible

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geodesics) if any isometric map from an interval in R to X extends to at least one isometricmap from R to X. We will put a special emphasis on the case when X is a (proper, geodesi-cally complete) R-tree, that is, a uniquely arcwise connected geodesic metric space. In theIntroduction, we have denoted by Y the geodesically complete proper locally CATp´1q goodorbispace ΓzX, see for instance [GH, Ch. 11] for the terminology.

We denote by B8X the space at infinity of X, which consists of the asymptotic classes ofgeodesic rays in X. It coincides with the space of (Freudenthal’s) ends of X when X is anR-tree. We denote by ΛΓ the limit set of Γ and by C ΛΓ the convex hull in X of ΛΓ.

When X is an R-tree, then a subset D of X is convex if and only if it is connected, andwe will call it a subtree. In particular, if X is an R-tree, then C ΛΓ is equal to the union ofthe geodesic lines between pairs of distinct points in ΛΓ, since this union is connected andcontained in C ΛΓ.

A point ξ P B8X is called a conical limit point if there exists a sequence of orbit points ofx0 under Γ converging to ξ while staying at bounded distance from a geodesic ray ending atξ. The set of conical limit points is the conical limit set ΛcΓ of Γ.

A point p P ΛΓ is a bounded parabolic limit point of Γ if its stabiliser Γp in Γ acts properlydiscontinuously with compact quotient on ΛΓ ´ tpu. The discrete nonelementary group ofisometries Γ of X is said to be geometrically finite if every element of ΛΓ is either a conicallimit point or a bounded parabolic limit point of Γ (see for instance [Bowd], as well as [Pau2]when X is an R-tree, and [DaSU] for a very interesting study of equivalent conditions in aneven greater generality).

For all x P X YB8X and A Ă X, the shadow of A seen from x is the subset OxA of B8Xconsisting of the positive endpoints of the geodesic rays starting at x and meeting A if x P X,and of the geodesic lines starting at x and meeting A if x P B8X.

We denote by IsompXq the isometry group of X. The translation length of an isometryγ P IsompXq is

λpγq “ infxPX

dpx, γxq .

Recall that γ P IsompXq is loxodromic if λpγq ą 0, and that then

Axγ “ tx P X : dpx, γxq “ λpγqu

is (the image of) a geodesic line in X, called the translation axis of γ.We will need the following well-known lemma later on. An element of Γ is primitive if

there is no γ0 P Γ and k P N´ t0, 1u such that γ “ γk0 .

Lemma 2.1. (1) For every loxodromic element γ P Γ, there exists a primitive loxodromicelement γ0 P Γ and k P N´ t0u such that γ “ γk0 .

(2) If α, γ P Γ are loxodromic with Axγ “ Axα, then there exists p, q P Z and a primitiveloxodromic element γ0 P Γ such that α “ γp0 and γ “ γq0.

(3) For all α, γ P Γ such that α is loxodromic, if γ centralises1 α, then either γ pointwisefixes the translation axis of α, or γ is loxodromic, with Axγ “ Axα.

(4) For every loxodromic element γ P Γ, for all A ą 0 and r ą 0, there exists L ą 0 suchthat for every loxodromic element α P Γ, if λpαq “ λpγq ď A and if Axγ and Axα havesegments of length at least L at Hausdorff distance at most r, then Axγ “ Axα.

1that is, commutes with α

26 19/12/2016

Proof. We only give a proof of Claim (4) and refer to for instance [BridH] for proofs of thefirst three classical assertions.

Since the action of Γ on X is properly discontinuous, and by the compactness of γZzAxγ ,there exists N ě 1 such that for all x P Axγ , the cardinality of tβ P Γ : dpx, βxq ď 2ruis at most N . Let L “ AN . For every loxodromic element α P Γ with λpαq “ λpγq ď A,assume that rx, ys and rx1, y1s are segments in Axγ and Axα respectively, with length exactlyL such that dpx, x1q, dpy, y1q ď r. We may assume, up to replacing them by their inverses,that γ translates from x towards y and α translates from x1 towards y1. In particular fork “ 0, . . . , N , we have dpα´kγkx, xq ď dpγkx, αkx1q ` dpx1, xq ď 2r since γkx and αkx1 arerespectively the points at distance kλpγq ď kA ď L from x and x1 on the segments rx, ysand rx1, y1s. Hence by the definition of N , there exists k ‰ k1 such that α´kγk “ α´k

1

γk1 .

Therefore γk´k1 “ αk´k1 , which implies by Assertion (3) that Axγ “ Axα. l

For every x P X, recall that the Gromov-Bourdon visual distance dx on B8X seen from x(see [Bou]) is defined by

dxpξ, ηq “ limtÑ`8

e12pdpξt, ηtq´dpx, ξtq´dpx, ηtqq , (2.1)

where ξ, η P B8X and t ÞÑ ξt, ηt are any geodesic rays ending at ξ, η respectively. By thetriangle inequality, for all x, y P ĂM and ξ, η P B8ĂM , we have

e´dpx, yq ďdxpξ, ηq

dypξ, ηqď edpx, yq . (2.2)

In particular, the identity map from pB8X, dxq to pB8X, dyq is a bilipschitz homeomorphism.Under our assumptions, pB8X, dx0q is hence a compact metric space, on which IsompXq actsby bilipschitz homeomorphisms. The following well-known result compares shadows of ballsto balls for the visual distance.

Lemma 2.2. For every geodesic ray ρ in X, starting from x P X and ending at ξ P B8X, forall R ě 0 and t P sR,`8r , we have

Bdxpξ,R e´tq Ă OxBpρptq, Rq Ă Bdxpξ, e

R e´tq .

Proof. The lower bound is for instance the lower bound in [HeP2, Lem. 3.1] (which onlyuses the CATp´1q property). In order to prove the upper bound, let ξ1 P OxBpρptq, Rq andlet ρ1 be the geodesic ray from x to ξ1. The closest point p to ρptq on the image of ρ1 satisfiesdpp, ρptqq ď R, hence dpx, pq ě t´R. Then

dxpξ1, ξq ď lim sup

t1Ñ`8e

12p dpρpt1q, ρptqq`dpρptq, pq`dpp, ρ1ptqq q´t1

ď limt1Ñ`8

e12ppt1´tq`R`pt1´t`Rqq´t1 “ eRe´t .

Therefore ξ1 P Bdxpξ, eR e´tq. l

The Busemann cocycle of X is the map β : B8X ˆX ˆX Ñ R defined by

pξ, x, yq ÞÑ βξpx, yq “ limtÑ`8

dpξt, xq ´ dpξt, yq ,

27 19/12/2016

where t ÞÑ ξt is any geodesic ray ending at ξ. The above limit exists and is independent ofx0, and we have

|βξpx, yq | ď dpx, yq . (2.3)

The horosphere with centre ξ P B8X through x P X is ty P X : βξpx, yq “ 0u, andty P X : βξpx, yq ď 0u is the (closed) horoball centred at ξ bounded by this horosphere.Horoballs are nonempty proper closed (strictly) convex subsets of X. Given a horoball Hand t ě 0, we denote by H rts “ tx P H : dpx, BH q ě tu the horoball contained in H(hence centred at the same point at infinity as H ) whose boundary is at distance t from theboundary of H

2.3 Generalised geodesic lines

Letp

GX be the space of 1-Lipschitz maps w : RÑ X which are isometric on a closed intervaland locally constant outside it.2 This space has been introduced by Bartels and Lück in[BartL], to which we refer for the following basic properties. The elements of

p

GX are calledthe generalised geodesic lines of X. We endow

p

GX with the distance d “ dp

GXdefined by

@ w,w1 P

p

GX, dpw,w1q “

ż `8

´8

dpwptq, w1ptqq e´2|t| dt . (2.4)

The group IsompXq acts isometrically onp

GX by postcomposition. The distance d inducesthe topology of uniform convergence on compact subsets on

p

GX, andp

GX is a proper metricspace.

The geodesic flow pgtqtPR onp

GX is the one-parameter group of homeomorphisms of thespace

p

GX defined by gtw : s ÞÑ wps ` tq for all w Pp

GX and t P R. It commutes with theaction of IsompXq. If w is isometric exactly on the interval I, then g´tw is isometric exactlyon the interval t` I. Note that for all w P

p

GX and s P R, we have

dpw, gswq ď |s| , (2.5)

with equality if w P GX.The footpoint projection is the IsompXq-equivariant 1

2 -Hölder-continuous map π :

p

GX Ñ

X defined by πpwq “ wp0q for all w P

p

GX. The antipodal map ofp

GX is the IsompXq-equivariant isometric map ι :

p

GX Ñ

p

GX defined by ιw : s ÞÑ wp´sq for all w Pp

GX, whichsatisfies ι ˝ gt “ g´t ˝ ι for every t P R and π ˝ ι “ π.

The positive and negative endpoint maps are the continuous maps fromp

GX to X Y B8Xdefined by

w ÞÑ w˘ “ limtÑ˘8

wptq .

The space GX of geodesic lines in X is the IsompXq-invariant closed subspace ofp

GX

consisting of the elements ` Pp

GX with `˘ P B8X. Note that the distances on GX consideredin [BartL] and [PauPS] are topologically equivalent, although slightly different from the re-striction to GX of the distance defined in Equation (2.4). The factor e´2|t| in this equation,

2that is, constant on each complementary component

28 19/12/2016

sufficient in order to deal with Hölder-continuity issues, is replaced by e´t2?π in [PauPS]

and by e´|t|2 in [BartL] (so that the above 12 -Hölder-continuity claim of π does follow from

the one in [BartL]).We will also consider the IsompXq-invariant closed subspaces

G˘X “ tw P

p

GX : w˘ P B8Xu ,

and their IsompXq-invariant closed subspaces G˘, 0X consisting of the elements ρ P G˘Xwhich are isometric exactly on ˘r0,`8r.

The subspaces GX and G˘X satisfy G´X XG`X “ GX and they are invariant under thegeodesic flow. The antipodal map ι preserves GX, and maps G˘X to G¯X as well as G˘, 0X

to G¯, 0X. We denote again by ι : Γz

p

GX Ñ Γz

p

GX and by gt : Γz

p

GX Ñ Γz

p

GX the quotientmaps of ι and gt, for every t P R.

Let w P

p

GX be isometric exactly on an interval I of R. If I is compact then w is a(generalised) geodesic segment, and if I “ s´8, as or I “ ra,`8r for some a P R, then wis a (generalised) (negative or positive) geodesic ray in X. Any geodesic line pw P GX suchthat pw|I “ w|I is an extension of w. Note that pw is an extension of w if and only if γ pw is anextension of γw for any γ P IsompXq, if and only if ι pw is an extension of ιw, and if and onlyif gs pw is an extension of gsw for any s P R. For any subset Ω of

p

GX and any subset A of R,let

Ω|A “ tw|A : w P Ωu .

Remark 2.3. Let p`iqiPN be a sequence of generalised geodesic lines such that rt´i , t`i s is

the maximal segment on which `i is isometric. Let psiqiPN be a sequence in R such thatt˘i ´ si Ñ ˘8 as i Ñ `8 and `ipsiq stays in a compact subset of X, then dp`i,GXq Ñ 0as i Ñ `8. Furthermore if psiqiPN is bounded, then up to extracting a subsequence, p`iqiPNconverges to an element in GX.

This conceptually important observation explains how it is conceivable that long commonperpendicular segments may equidistribute towards measures supported on geodesic lines. SeeChapter 11 for further developments of these ideas.

2.4 The unit tangent bundle

In this work, we define the unit tangent bundle T 1X of X as the space of germs at 0 of thegeodesic lines in X, that is the quotient space

T 1X “ GX „

where ` „ `1 if and only if there exists ε ą 0 such that `|r´ε,εs “ `1|r´ε,εs. The canonicalprojection from GX to T 1X will be denoted by ` ÞÑ v`. When X is a Riemannian manifold,the spaces GX and T 1X canonically identify with the usual unit tangent bundle of X, but ingeneral, the map ` ÞÑ v` has infinite fibers.

We endow T 1X with the quotient distance d “ dT 1X of the distance of GX, defined by:

@ v, v1 P T 1X, dT 1Xpv, v1q “ inf

`, `1PGX : v“v`, v1“v`1dp`, `1q . (2.6)

29 19/12/2016

It is easy to check that this distance is indeed Hausdorff, hence that T 1X is locally compact,and that it induces on T 1X the quotient topology of the compact-open topology of GX. Themap ` ÞÑ v` is 1-Lipschitz.

The action of IsompXq on GX induces an isometric action of IsompXq on T 1X. Theantipodal map and the footpoint projection restricted to GX respectively induce an IsompXq-equivariant isometric map ι : T 1X Ñ T 1X and an IsompXq-equivariant 1

2 -Hölder-continuousmap π : T 1X Ñ X called the antipodal map and footpoint projection of T 1X. The canonicalprojection from GX to T 1X is IsompXq-equivariant and commutes with the antipodal map:For all γ P IsompXq and ` P GX, we have γv` “ vγ`, ιv` “ vι` and πpv`q “ πp`q. We denoteagain by ι : ΓzT 1X Ñ ΓzT 1X the quotient map of ι.

Let B28X be the subset of B8XˆB8X which consists of pairs of distinct points at infinity of

X. Hopf’s parametrisation of GX is the homeomorphism which identifies GX with B28XˆR,

by the map ` ÞÑ p`´, ``, tq, where t is the signed distance from the closest point to thebasepoint x0 on the geodesic line ` to `p0q.3 We have gsp`´, ``, tq “ p`´, ``, t ` sq for alls P R, and for all γ P Γ, we have γp`´, ``, tq “ pγ`´, γ``, t ` tγ, `´, ``q where tγ, `´, `` P Rdepends only on γ, `´ and ``. In Hopf’s parametrisation, the restriction of the antipodalmap to GX is the map p`´, ``, tq ÞÑ p``, `´,´tq.

The strong stable leaf of w P G`X is

W`pwq “

` P GX : limtÑ`8

dp`ptq, wptqq “ 0(

,

and the strong unstable leaf of w P G´X is

W´pwq “ ιW`pιwq “

` P GX : limtÑ´8

dp`ptq, wptqq “ 0(

.

For every w P G˘X, let dW˘pwq be Hamenstädt’s distance on W˘pwq defined as follows:4 for

all `, `1 PW˘pwq, letdW˘pwqp`, `

1q “ limtÑ`8

e12dp`p¯tq, `1p¯tqq´t .

The above limits exist, and Hamenstädt’s distances are distances inducing the original topol-ogy on W˘pwq. For all `, `1 PW˘pwq and γ P IsompXq, we have

dW˘pγwqpγ`, γ`1q “ dW˘pwqp`, `

1q “ dW¯pιwqpι`, ι`1q .

Furthermore, for every s P R, we have for all `, `1 PW˘pwq

dW˘pgswqpgs`, gs`1q “ e¯sdW˘pwqp`, `

1q . (2.7)

If X is an R-tree, for all w P G`X and `, `1 P W`pwq, if rs,`8r is the maximal intervalon which ` and `1 agree, then dW`pwqp`, `

1q “ es.The following lemma compares the distance in GX with Hamenstädt’s distance for two

leaves in the same strong (un)stable leaf.

Lemma 2.4. There exists a universal constant c ą 0 such that for all w P G˘X and `, `1 PW˘pwq, wa have

dp`, `1q ď c dW˘pwqp`, `1q .

3More precisely, `ptq is the closest point to x0 on `.4See [HeP1, Appendix] and compare with [Ham1].

30 19/12/2016

Proof. We refer to [PaP14a, Lem. 3] for a proof of this result. Note that the distance onGX considered in loc. cit. is slightly different from the one in this text, but the proof adaptseasily. l

Let H be a horoball in X, centred at ξ P B8X. The strong stable leaves W`pwq areequal for all geodesic rays w starting at time t “ 0 from a point of BH and converging toξ. Using the homeomorphism ` ÞÑ `´ from W`pwq to B8X ´ tξu, Hamenstädt’s distance onW`pwq defines a distance dH on B8X ´ tξu that we also call Hamenstädt’s distance. For all`, `1 PW`pwq, we have

dH p`´, ``q “ dW`pwqp`, `1q ,

and for all η, η1 P B8X ´ tξu, we have

dH pη, η1q “ lim

tÑ`8e

12dp`ηp´tq, `η1 p´tqq´t , (2.8)

where `η, `η1 are the geodesic lines starting from η, η1 respectively, ending at ξ, and passingthrough the boundary of H at time t “ 0. Note that for every t ě 0, if H rts is the horoballcontained in H whose boundary is at distance t from the boundary of H , then we have

dH rts “ e´t dH . (2.9)

Let w P G˘X and η1 ą 0. We define B˘pw, η1q as the set of ` P W˘pwq such that thereexists an extension pw P GX of w with dW˘pwqp`, pwq ă η1. In particular, B˘pw, η1q containsall the extensions of w, and is the union of the open balls centred at the extensions of w, ofradius η1, for Hamenstädt’s distance on W˘pwq.

The union over t P R of the images under gt of the strong stable leaf of w P G`X is thestable leaf

W 0`pwq “ď

tPRgtW`pwq

of w, which consists of the elements ` P GX with `` “ w`. Similarly, the unstable leaf ofw P G´X

W 0´pwq “ď

tPRgtW´pwq ,

consists of the elements ` P GX with `´ “ w´. Note that the (strong) (un)stable leaves aresubsets of the space of geodesic lines GX.

The unstable horosphere H´pwq of w P G´X is the horosphere in X centred at w´ andpassing through pwp0q for any extension pw P GX of w. The stable horosphere H`pwq ofw P G`X is the horosphere in X centred at w` and passing through pwp0q for any extensionpw P GX of w. These horospheres H˘pwq do not depend on the chosen extensions pw ofw P G˘X. The unstable horoball HB´pwq of w P G´X and stable horoball HB`pwq ofw P G`X are the horoballs bounded by these horospheres. Note that πpW˘pwqq “ H˘pwqfor every w P G˘X, and that wp0q belongs to H˘pwq if and only if w is isometric at least on˘r0,`8r.

31 19/12/2016

2.5 Normal bundles and dynamical neighbourhoods

In this Section, adapting [PaP16c, §2.2] to the present context, we define spaces of geodesicrays that generalise the unit normal bundles of submanifolds of negatively curved Riemannianmanifolds. When X is a manifold, these normal bundles are submanifolds of the unit tangentbundle of X, which identifies with GX. In general and in particular in trees, it is essential touse geodesic rays to define normal bundles, and not geodesic lines.

Let D be a nonempty proper (that is, different from X) closed convex subset in X. Wedenote by BD its boundary in X and by B8D its set of points at infinity. Let

PD : X Y pB8X ´ B8Dq Ñ D

be the (continuous) closest point map to D, defined on ξ P B8X ´ B8D by setting PDpξq tobe the unique point in D that minimises the function y ÞÑ βξpy, x0q from D to R. The outerunit normal bundle B1

`D of (the boundary of) D is

B1`D “ tρ P G`, 0X : PDpρ`q “ ρp0qu .

The inner unit normal bundle B1´D of (the boundary of) D is

B1´D “ ιB1

`D “ tρ P G´, 0X : PDpρ´q “ ρp0qu.

Note that B1`D and B1

´D are spaces of geodesic rays. If X is a smooth manifold, then thesespaces have a natural identification with subsets of GX because every geodesic ray is therestriction of a unique geodesic line. But this does not hold in general.

Remark 2.5. AsX is assumed to be proper with extendible geodesics, we have πpB1˘Dq “ BD.

To see this, let x P BD and let pxkqkPN be a sequence of points in the complement of Dconverging to x. For all k P N, let ρk P B1

`D be a geodesic ray with ρkp0q “ PDpxkq and suchthat the image of ρk contains xk. As the closest point map does not increase distances, thesequence pPDpxkqqkPN converges to x. Since X is proper, the space B8X is compact and thesequence ppρkq`qkPN has a subsequence that converges to a point ξ P B8X. The claim followsfrom the continuity of the closest point map.

The failure of the equality when X is not proper is easy to see for example, when X isthe R-tree constructed by starting with the Euclidean line D “ R and attaching a copy of thehalfline r0,`8r to each x P D such that x ą 0. Then 0 P BD ´ πpB1

˘Dq.

The restriction of the positive (respectively negative) endpoint map to B1`D (respectively

B1´D) is a homeomorphism to its image B8X´B8D. We denote its inverse map by P˘D . Notethat PD “ π˝P˘D . For every isometry γ of X, we have B1

˘pγDq “ γ B1˘D and P˘γD ˝γ “ γ˝P˘D .

In particular, B1˘D is invariant under the isometries of X that preserve D.

For every w P G˘X, we have a canonical homeomorphism N˘w : W˘pwq Ñ B1¯HB˘pwq,

that associates to each geodesic line ` PW˘pwq the unique geodesic ray ρ P B1¯HB˘pwq such

that `¯ “ ρ¯, or, equivalently, such that `ptq “ ρptq for every t P R with ¯t ą 0. It is easy tocheck that N˘γw ˝ γ “ γ ˝N˘w for every γ P IsompXq.

We defineU ˘D “ t` P GX : `˘ R B8Du . (2.10)

Note that U ˘D is an open subset of GX, invariant under the geodesic flow. We have U ˘

γD “

γU ˘D for every isometry γ of X and, in particular, U ˘

D is invariant under the isometries of X

32 19/12/2016

preserving D. Define f˘D : U ˘D Ñ B1

˘D as the composition of the continuous endpoint map` ÞÑ `˘ from U ˘

D onto B8X ´ B8D and the homeomorphism P˘D from B8X ´ B8D to B1˘D.

The continuous map f˘D takes ` P U ˘D to the unique element ρ P B1

˘D such that ρ˘ “ `˘.The fiber of ρ P B1

`D for f`D is exactly the stable leaf W 0`pρq, and the fiber of ρ P B1´D for

f´D is the unstable leaf W 0´pρq. For all γ P IsompXq and t P R, we have

f˘γD ˝ γ “ γ ˝ f˘D and f˘D ˝ gt “ f˘D . (2.11)

Let w P G˘X and η, η1 ą 0. We define the dynamical pη, η1q-neighbourhood of w by

V ˘w, η, η1 “ď

sPs´η, η r

gsB˘pw, η1q . (2.12)

Example 2.6. If X is an R-tree, w P G`X and η ă ln η1, then V `w, η, η1 is as in the followingpicture.

πpV `w, η, η1q

w

πpB`pw, η1qq

pwpln η1qpwp0q

w` “ pw`

Clearly, B˘pw, η1q “ ιB¯pιw, η1q, and hence we have V ˘w, η, η1 “ ιV ¯ιw, η, η1 . Furthermore, forevery s P R,

gsB˘pw, η1q “ B˘pgsw, e¯sη1q hence gsV ˘w, η, η1 “ V ˘gsw, η, e¯sη1

. (2.13)

For every γ P IsompXq, we have γB˘pw, η1q “ B˘pγw, η1q and γV ˘w, η, η1 “ V ˘γw, η, η1 . The mapfrom s´η, ηr ˆB˘pw, η1q to V ˘w, η, η1 defined by ps, `1q ÞÑ gs`1 is a homeomorphism.

For all subsets Ω´ of G`X and Ω` of G´X, let

V ˘η, η1pΩ

¯q “ď

wPΩ¯

V ˘w, η, η1 , (2.14)

that we call the dynamical neighbourhoods of Ω¯. Note that they are subsets of GX, not ofG˘X. The families pV ˘

η, η1pΩ¯qqη,η1ą0 are nondecreasing in η and in η1. For every γ P IsompXq,

we have γV ˘η, η1pΩ

¯q “ V ˘η, η1pγΩ¯q and for every t ě 0, we have

g˘tV ˘η, η1pΩ

¯q “ V ˘

η, e´tη1pg˘tΩ¯q . (2.15)

Note thatď

η , η1ą0

V ˘η, η1pB

1˘Dq “ U ˘

D ,

and thatŞ

η, η1ą0 V ˘η, η1pB

1˘Dq is the set of all extensions in GX of the elements of B1

˘D. Assumethat Ω¯ is a subset of B1

˘D. The restriction of f˘D to V ˘η, η1pΩ

¯q is a continuous map onto Ω¯,with fiber over w P Ω¯ the open subset V ˘w, η, η1 of W

0˘pwq.

We will need the following elementary lemma in Section 10.4.

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Lemma 2.7. There exists a universal constant c1 ą 0 such that for every w P G`X which isisometric on rsw,`8r and every ` P V `w, η, η1 , we have

dp`, wq ď c1pη ` η1 ` eswq .

Proof. By Equation (2.12) and by the definition of B`pw, η1q in Section 2.3, there exists P s ´ η,`ηr and an extension pw P GX of w such that dW`pwqpg

s`, pwq ď η1. By Equation(2.5), we have dp`, gs`q ď |s| ď η. By Lemma 2.4, we have dpgs`, pwq ď c dW`pwqpg

s`, pwq ď c η1.

By the definition of the distance onp

GX (see Equation (2.4)), we have

dp pw,wq ď

ż sw

´8

|sw ´ t| e´2|t| dt “ Opeswq .

Therefore the result follows from the triangle inequality

dp`, wq ď dp`, gs`q ` dpgs`, pwq ` dp pw,wq . l

2.6 Creating common perpendiculars

Let D´ and D` be two nonempty proper closed convex subsets of X, where X is as inthe beginning of Section 2.2. A geodesic arc α : r0, T s Ñ X, where T ą 0, is a commonperpendicular of length T from D´ to D` if there exist w¯ P B1

˘D¯ such that w´|r0, T s “

g´Tw`|r0, T s “ α. Since X is CATp´1q, this geodesic arc α is the unique shortest geodesicsegment from a point of D´ to a point of D`. There is a common perpendicular from D´ toD` if and only if the closures of D´ and D` in X YB8X are disjoint. When X is an R-tree,then two closed subtrees of X have a common perpendicular if and only if they are nonemptyand disjoint.

One of the aims of this text is to count orbits of common perpendiculars between twoequivariant families of closed convex subsets of X. The crucial remark is that two nonemptyproper closed convex subsets D´ and D` of X have a common perpendicular α of lengtha given T ą 0 if and only if the subsets gT 2B1

`D´|r´T

2, T

2sand g´T 2B1

´D`|r´T

2,T

2sof

p

GX

intersect. This intersection then consists of the common perpendicular from D´ to D`

reparametrised by r´T2 ,

T2 s. As a controlled perturbation of this remark, we now give an

effective creation result of common perpendiculars in R-trees. It has a version satisfied for Xin the generality of Section 2.2, see the end of this Section.

Lemma 2.8. Assume that X is an R-tree. For all R ą 1, η P s0, 1s and t ě 2 lnR ` 4,for all nonempty closed connected subsets D´, D` in X, and for every geodesic line ` Pgt2V `

η,RpB1`D

´q X g´t2V ´η,RpB

1´D

`q, there exist s P s´2η, 2ηr and a common perpendicular rcfrom D´ to D` such that• the length of rc is t` s,• the endpoint of rc in D¯ is w¯p0q where w¯ “ f˘

D¯p`q,

• the footpoint `p0q of ` lies on rc, and

max!

dpw´` t

2

˘

, `p0qq, dpw``

´t

2

˘

, `p0qq)

ď η .

34 19/12/2016

`p0q

w´p t2q w`p´ t2q

ď lnR ď lnR

x´ x`

D´

w`p0q

D`

`p t2 ` s`q

w´p0q

`p´ t2 ´ s

´q

Proof. Let R, η, t,D˘, ` be as in the statement. By the definition of the sets V ¯η,RpB

1¯D

˘q,there exist geodesic rays w˘ P B1

¯D˘, geodesic lines pw˘ P GX extending w˘, and s˘ P

s´η,`ηr, such that `˘ “ pw¯q˘ and

dW˘pw¯qpg¯ t

2¯s¯`, pw¯q ď R .

Let x˘ be the closest point to w˘p0q on `. By the definition of Hamenstädt’s distances, wehave

dpw˘p0q, x˘q “ dp`p˘t

2˘ s˘q, x˘q ď lnR ,

and in particular x˘ “ w˘p0q if and only if `p˘ t2 ˘ s˘q “ w˘p0q. As t ě 2 lnR ` 4 and

|s˘| ď 2η ď 2, the points `p´ t2 ´ s´q, x´, `p0q, x`, `p t2 ` s`q are in this order on `. In

particular, the segment rw´p0q, x´s Y rx´, x`s Y rx`, w`p0qs is a nontrivial geodesic segmentfrom a point of D´ to a point of D` that meets D¯ only at an endpoint. Hence, D´ and D`

are disjoint, and rw´p0q, w`p0qs is the image of the common perpendicular from D´ to D`.Let s “ s´ ` s`. The length of rc is p t2 ` s

`q ´ p´ t2 ´ s

´q “ t` s. The point `p0q lies onrc, we have w¯ “ f˘

D¯p`q and the endpoints of rc are w˘p0q. Furthermore,

dpw¯p˘t

2q, `p0qq “

ˇ

ˇ dpw¯p˘t

2q, w¯p0qq ´ dp`p0q, `p¯

t

2¯ s¯qq

ˇ

ˇ “ |s¯| ď η . l

When X is as in the beginning of Section 2.2, the statement and the proof of the followinganalog of Lemma 2.8 is slightly more technical. We refer to [PaP16c, Lem. 7] for a proof inthe Riemannian case, and we leave the extension to the reader, since we will not need it inthis book.

Lemma 2.9. Let X be as in the beginning of Section 2.2. For every R ą 0, there existt0, c0 ą 0 such that for all η P s0, 1s and all t P rt0,`8r , for all nonempty closed convexsubsets D´, D` in X, and for all w P gt2V `

η,RpB1`D

´q X g´t2V ´η,RpB

1´D

`q, there exist s Ps ´ 2η, 2ηr and a common perpendicular rc from D´ to D` such that• the length of rc is contained in rt` s´ c0 e

´ t2 , t` s` c0 e

´ t2 s,

• if w¯ “ f˘D¯pwq and if p˘ is the endpoint of rc in D˘, then dpπpw˘q, p˘q ď c0 e

´ t2 ,

• the basepoint πpwq of w is at distance at most c0 e´ t

2 from a point of rc, and

maxt dpπpgt2w´q, πpwqq, dpπpg´

t2w`q, πpwqq u ď η ` c0 e

´ t2 . l

2.7 Metric and simplicial trees, and graphs of groups

Metric and simplicial trees and graphs of groups are important examples throughout the restof this work. In this Section, we recall the definitions and basic properties of these objects.

35 19/12/2016

We use Serre’s definitions in [Ser3, §2.1] concerning a graph X, with V X and EX its set ofvertices and edges, and opeq, tpeq and e the initial vertex, terminal vertex and opposite edgeof an edge e P EX. Recall that a connected graph is bipartite if it is endowed with a partitionof its set of vertices into two nonempty subsets such that any two elements of either subsetare not related by an edge.

The degree of a vertex x P V X is the cardinality of the set te P EX : opeq “ xu. For allj, k P N, a graph X is k-regular if the degree of each vertex x P V X is k, and it is pj, kq-biregularif it is bipartite with the partition of its vertices into the two subsets consisting of verticeswith degree j and with degree k respectively.

A metric graph pX, λq is a pair consisting of a graph X and a map λ : EXÑ s0,`8r witha positive lower bound5 such that λpeq “ λpeq, called its edge length map. A simplicial graphX is a metric graph whose edge length map is constant equal to 1.

The topological realisation of a graph X is the topological space obtained from the collectionpIeqePEX of closed unit intervals Ie by the finest equivalence relation that identifies intervalscorresponding to an edge and its opposite edge by the map t ÞÑ 1´ t and identifies the originsof the intervals Ie1 and Ie2 if and only if ope1q “ ope2q, see [Ser3, Sect. 2.1].

The geometric realisation of a metric tree pX, λq is the topological realisation of X endowedwith the maximal geodesic metric that gives length λpeq to the topological realisation of eachedge e P EX, and we denote it by X “ |X|λ. We identify V X with its image in X. The metricspace X determines pX, λq up to subdivision of edges, hence we will often not make a strictdistinction between X and pX, λq. In particular, we will refer to convex subsets of pX, λq asconvex subsets of X, etc.

If X is a tree, the metric space X is an R-tree, hence it is a CATp´1q space. Since λ isbounded from below by a positive constant, the R-tree X is geodesically complete if and onlyif X has no terminal vertex (that is, no vertex of degree 1).

We will denote by AutpX, λq, and AutX in the simplicial case, the group of edge-preservingisometries ofX that have no inversions.6 Since the edge length map has a positive lower bound,the metric space X is proper if and only if X is locally finite. In this case, the nonelementarydiscrete subgroups Γ of isometries of X we will consider will always be edge-preserving andwithout inversion.

A locally finite metric tree pX1, λq is uniform if there exists some discrete subgroup Γ1 ofAutpX1, λq such that Γ1zX1 is a finite graph. See [BasK, BasL] for characterisations of thisproperty in the case of simplicial trees.

The space of generalised discrete geodesic lines of a locally finite simplicial tree X is thelocally compact space

p

GX of 1-Lipschitz mappings w from R to its geometric realisationX “ |X|1 which are isometric on a closed interval with endpoints in Z Y t´8,`8u andlocally constant outside it, such that wp0q P V X (or equivalently wpZq Ă V X). Note that

p

GXis hence a proper subset of

p

GX.By restriction to

p

GX, or intersection withp

GX, of the objects defined in Sections 2.3 and2.5 for

p

GX, we define the distance d onp

GX, the subspaces G˘X, GX, G˘,0X, the strongstable/unstable leaves W˘pwq of w P G˘X and their Hamenstädt distances dW˘pwq, thestable/unstable leaves W 0˘pwq of w P G˘X, the outer and inner unit normal bundles B1

˘Dof a nonempty proper simplicial subtree D of X, the dynamical neighbourhoods V ˘

η, η1pΩ¯q of

5This assumption, though not necessary at this stage, will be used repeatedly in this text, hence we preferto add it to the definition.

6An automorphism g of a graph has an inversion if there exists an edge e of the graph such that ge “ e.

36 19/12/2016

subsets Ω¯ of B1˘D as well as the fibrations

f˘D : U ˘D “ t` P GX : `˘ R B8Du Ñ B1

˘D ,

whose fiber over ρ P B1˘D is W 0˘pρq. Note that some definitions actually simplify when

considering generalised discrete geodesic lines. For instance, for all w P G˘X, η1 ą 0 and η Ps0, 1r , the dynamical neighbourhood V ˘w, η, η1 is equal to B

˘pw, η1q, and is hence independentof η P s0, 1r .

Besides the map π : GX Ñ V X defined as in the continuous case by ` ÞÑ `p0q, we haveanother natural map Tπ : GX Ñ EX, which associates to ` the edge e with opeq “ `p0q andtpeq “ `p1q. This map is equivariant under the group of automorphisms (without inversions)AutpXq of X, and we also denote by Tπ : ΓzGXÑ ΓzEX its quotient map, for every subgroupΓ of AutpXq.

If X has no terminal vertex, for every e P EX, let

BeX “ t`` : ` P GX, Tπp`q “ eu

be the set of points at infinity of the geodesic rays whose initial (oriented) edge is e.Given x0 P V X, the discrete Hopf parametrisation now identifies GX with B2

8XˆZ by themap ` ÞÑ p`´, ``, tq where t P Z is the signed distance from the closest vertex to the basepointx0 on the geodesic line ` to the vertex `p0q.

The discrete time geodesic flow pgtqtPZ onp

GX is the one-(discrete-)parameter group ofhomeomorphisms of

p

GX consisting of (the restriction top

GX of) the integral time maps of thecontinuous time geodesic flow of the geometric realisation of X: we have gtw : s ÞÑ wps ` tq

for all w Pp

GX and t P Z.

Recall (see for instance [Ser3, BasL]) that a graph of groups pY, G˚q consists of

‚ a graph Y, which is connected unless otherwise stated,

‚ a group Gv for every vertex v P V Y,

‚ a group Ge for every edge e P EY such that Ge “ Ge,

‚ an injective group morphism ρe : Ge Ñ Gtpeq for every edge e P V Y.

We will identify Ge with its image in Gtpeq by ρe, unless the meaning is not clear (which mightbe the case for instance if opeq “ tpeq).

A subgraph of subgroups of pY, G˚q is a graph of groups pY1, G1˚q where‚ Y1 is a subgraph of Y,‚ for every v P V Y1, the group G1v is a subgroup of Gv,‚ for every e P EY, the group G1e is a subgroup of Ge, and the monomorphism G1e Ñ G1tpeq

is the restriction of the monomorphism Ge Ñ Gtpeq, and

G1tpeq X ρepGeq “ ρepG1eq .

This condition, first introduced in [Bass, Coro. 1.14], is equivalent to the injectivity of thenatural map G1tpeqρepG

1eq Ñ GtpeqρepGeq. It implies by [Bass, 2.15] that when the underlying

basepoint is chosen in Y1, the fundamental group of pY1, G1˚q injects in the fundamental group

37 19/12/2016

of pY, G˚q, and the Bass-Serre tree X1 of pY1, G1˚q injects in an equivariant way in the Bass-Serre tree X of pY, G˚q.

Note that the fundamental group of pY, G˚q does not always act faithfully on its Bass-Serretree X, that is, the kernel of its action might be nontrivial.

The edge-indexed graph pY, iq of the graph of groups pY, G˚q is the graph Y endowed withthe map i : EYÑ N´ t0u defined by ipeq “ rGopeq : Ges (see for instance [BasK, BasL]).

In Section 12.4, we will consider metric graphs of groups pY, G˚, λq which are graphs ofgroups endowed with an edge length function λ : EY Ñ s0,`8r (with λpeq “ λpeq for everye P EY).

Example 2.10. The main examples of graphs of groups that we will consider in this textare the following ones. Let X be a simplicial tree and let Γ be a subgroup of AutpXq. Thequotient graph of groups ΓzzX is the following graph of groups pY, G˚q (having finite vertexgroups if X is locally finite and Γ is discrete). Its underlying graph Y is the quotient graphΓzX. Fix a lift rz P V XY EX for every z P V YY EY. For every e P EY, assume that re “ re,and fix an element ge P Γ such that ge Ątpeq “ tpreq. For every y P V Y Y EY, take as Gy thestabiliser Γ

ry in Γ of the fixed lift ry. Take as monomorphism ρe : Ge Ñ Gtpeq the restrictionto Γ

re of the conjugation γ ÞÑ g´1e γge by g´1

e .

The volume form of a graph of finite groups pY, G˚q is the measure volpY, G˚q on the discreteset V Y, such that for every y P V Y,

volpY, G˚qptyuq “1

|Gy|,

where |Gy| is the order of the finite group Gy. Its total mass, called the volume of pY, G˚q, is

VolpY, G˚q “ volpY, G˚q “ÿ

yPV Y

1

|Gy|.

We denote by L2pY, G˚q “ L2pV Y, volpY,G˚qq the Hilbert space of square integrable mapsV YÑ C for this measure volpY,G˚q, and by f ÞÑ f2 and pf, gq ÞÑ xf, gy2 its norm and scalarproduct. Let

L20pY, G˚q “ tf P L2pY, G˚q :

ż

f d volpY, G˚q “ 0u .

When VolpY, G˚q is finite, L20pY, G˚q is the orthogonal subspace to the constant functions.

We also consider a (edge-)volume form TvolpY, G˚q on the discrete set EY such that forevery e P EY,

TvolpY, G˚qpteuq “1

|Ge|,

with total massTVolpY, G˚q “ TvolpY, G˚q “

ÿ

ePEY

1

|Ge|.

The (edge-)volume form of a metric graph of groups pY, G˚, λq is given by

TvolpY, G˚, λq “ds

|Ge|

38 19/12/2016

on each edge e of Y parameterised by its arclength s, so that its total mass is

TVolpY, G˚, λq “ TvolpY, G˚,λq “ÿ

ePEY

λpeq

|Ge|.

For λ ” 1, this total mass agrees with that of the discrete definition above.

Remark 2.11. Note that TVolpY, G˚q “ CardpEYq when the edge groups are trivial. Wehave

TVolpY, G˚q “ÿ

ePEY

1

|Ge|“

ÿ

yPV Y

1

|Gy|

ÿ

ePEY, opeq“y

|Gy|

|Ge|“

ÿ

yPV Y

degpryq

|Gy|,

where ry is any lift of y in the Bass-Serre tree of pY, G˚q. In particular, if X is a uniformsimplicial tree and Γ is discrete subgroup of AutX, then the finiteness of VolpΓzzXq and ofTVolpΓzzXq are equivalent. Defining the volume form on V Y by tyu ÞÑ degpryq

|Gy |sometimes makes

formulas simpler, but we will follow the convention which occurs in the classical references(see for instance [BasL]).

If the Bass-Serre tree of pY, G˚q is pq ` 1q-regular, then

π˚TvolY, G˚ “ pq ` 1q volY, G˚ and TVolpY, G˚q “ pq ` 1qVolpY, G˚q . (2.16)

We say that a discrete group of isometries Γ of a locally finite metric or simplicial treepX, λq is a (tree) lattice of pX, λq if the quotient graph of groups ΓzzX has finite volume. If Xis simplicial, then this implies that Γ is a lattice in the locally compact group AutpXq (hencethat AutpXq is unimodular), the converse being true for instance if X is regular or biregular(see [BasK]). For instance, if Γ is a uniform lattice of X (or pX, λq), that is, if Γ is a discretesubgroup of AutpX, λq and if the quotient graph ΓzX is finite, then Γ is a lattice of pX, λq.

A graph of finite groups pY, G˚q is a cuspidal ray if Y is a simplicial ray such that thehomomorphisms Gen Ñ Gopenq are surjective for its sequence of consecutive edges peiqiPNoriented towards the unique end of Y. By [Pau1], a discrete group Γ1 of AutpXq (hence ofIsomp|X|1q) is geometrically finite if and only if it is nonelementary and if the quotient graph ofgroups by Γ1 of its minimal nonempty invariant subtree is the union of a finite graph of groupsand a finite number of cuspidal rays attached to the finite graph at their finite endpoints.

Remark 2.12. If X is a locally finite simplicial tree and if Γ1 is a geometrically finite discretegroup of AutpXq such that the convex hull of its limit set C ΛΓ1 is a uniform tree, then Γ1 isa lattice of C ΛΓ1.

Proof. Since C ΛΓ1 is uniform, there is a uniform upper bound on the length of an edgepath in C ΛΓ1 which injects in Γ1zC ΛΓ1 such that the stabiliser of each edge of this edge pathis equal to the stabilisers of both endpoints of this edge. It is hence easy to see that thevolume of each of the (finitely many) cuspidal rays in Γ1zzC ΛΓ1 is finite, by a geometric seriesargument. Hence the volume of Γ1zzC ΛΓ1 is finite. l

Note that contrarily to the case of Riemannian manifolds, there are many more (tree)lattices than there are geometrically finite (tree) lattices, even in regular trees, see for instance[BasL].

39 19/12/2016

In Part III of this text, we will consider simplicial graphs of groups that arise from arith-metics of non-Archimedean local fields. We say that Γ is algebraic if there exists a non-Archimedean local field pK (a finite extension of Qp for some prime p or the field of formalLaurent series over a finite field) and a connected semi-simple algebraic group G with finitecentre defined over pK, of pK-rank one, such that X identifies with the Bruhat-Tits tree of Gin such a way that Γ identifies with a lattice of Gp pKq. If Γ is algebraic, then Γ is geometri-cally finite by [Lub1]. Note that X is then bipartite, see Section 2 of op. cit. for a discussionand references. See Sections 14 and 15.1 for more details, and the subsequent Sections forarithmetic applications arising from algebraic lattices.

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Chapter 3

Potentials, critical exponents andGibbs cocycles

LetX be a geodesically complete proper CATp´1q space, let x0 P X be an arbitrary basepoint,and let Γ be a nonelementary discrete group of isometries of X.

In this Chapter, we define potentials on T 1X, which are new data on X in addition toits geometry. We introduce the fundamental tools associated with potentials, and we givesome of their basic properties. The development follows [PauPS] with modifications to fit thepresent more general context.

In Section 3.5, we introduce a natural method to associate a (Γ-invariant) potential rFc :T 1X Ñ R to a Γ-invariant function c : EX Ñ R defined on the set of edges of a simplicialor metric tree X, with geometric realisation X, that we call a system of conductances on X.This construction gives a nonsymmetric generalisation of electric networks.

3.1 Background on (uniformly local) Hölder-continuity

In this preliminary Section, we recall the notion of Hölder-continuity we will use in this text,which needs to be defined appropriately in order to deal with noncompactness issues. TheHölder-continuity will be used on one hand for potentials when X is a Riemannian manifoldin Section 3.2, and on the other hand for error term estimates in Chapters 9, 10 and 11.

As in [PauPS], we will use the following uniformly local definition of Hölder-continuousmaps. Let E and E1 be two metric spaces, and let α P s0, 1s. A map f : E Ñ E1 is• α-Hölder-continuous if there exist c, ε ą 0 such that for all x, y P E with dpx, yq ď ε, we

havedpfpxq, fpyqq ď c dpx, yqα .

• locally α-Hölder-continuous if for every x P E, there exists a neighbourhood U of x suchthat the restriction of f to U is α-Hölder-continuous;

• Hölder-continuous (respectively locally Hölder-continuous) if there exists α P s0, 1s suchthat f is α-Hölder-continuous (respectively locally α-Hölder-continuous);

• Lipschitz if it is 1-Hölder-continuous and locally Lipschitz if it is locally 1-Hölder-continuous.

Let E and E1 be two metric spaces. We say that a map f : E Ñ E1 has• at most linear growth if there exists a, b ě 0 such that dpfpxq, fpyqq ď a dpx, yq ` b for allx, y P E,

41 19/12/2016

• subexponential growth if for every a ą 0, there exists b ě 0 such that dpfpxq, fpyqq ďb ea dpx, yq for all x, y P E.

Remark 3.1. When E is a geodesic space, a consequence of the (uniformly local) Hölder-continuous property of f : E Ñ E1 is that f then has at most linear growth: the definitionimplies that dpfpxq, fpyqq ď c εα´1 dpx, yq ` c εα for all x, y in X, by subdividing the geodesicsegment in E from x to y in

Pdpx,yqε

T

segments of equal lengths at most ε and using the triangleinequality in E1.

For any metric space Z and α P s0, 1s, the Hölder norm of a bounded α-Hölder-continuousfunction f : Z Ñ R is

||f ||α “ f8 ` supx, yPZ

0ădpx,yqď1

|fpxq ´ fpyq|

dpx, yqα.

When the diameter of Z is bounded by 1,1 this coincides with the usual definition. Note thateven if the constant ε in the above definition of a α-Hölder-continuous map is less than 1, thisnorm is finite, since

supx, yPZ

εďdpx,yqď1

|fpxq ´ fpyq|

dpx, yqαď 2 ε´α f8 .

Note that for all bounded α-Hölder-continuous maps f, g : Z Ñ R, we have

fgα ď fα g8 ` f8 gα . (3.1)

We denote by C αc pZq (respectively C α

b pZq) the space of α-Hölder-continuous real-valuedfunctions with compact support (respectively which are bounded) on Z, endowed with thisnorm. Note that C α

b pZq is a Banach space.2

A stronger assumption than the Hölder regularity is the locally constant regularity, thatwe now define. Alhough it is only useful for totally disconnected metric spaces, several errorterms estimates in the literature use this stronger regularity (see for instance [AtGP, KemaPS]and Part III of this text).

Let ε ą 0. For every metric space E and every set E1, we say that a map f : E Ñ E1 isε-locally constant if f is constant on every closed ball of radius ε (or equivalently of radius atmost ε) in E. We say that f : E Ñ E1 is locally constant if there exists ε ą 0 such that f isε-locally constant.

Note that if E is a geodesic metric space and f : E Ñ E1 is locally constant, then f isconstant. But when E is for instance an ultrametric space, since two distinct closed ballsof the same radius are disjoint, the above definition turns out to be very interesting (andmuch used in representation theory in positive characteristic, for instance). For example, thecharacteristic function 1A of a subset A of E is ε-locally constant if and only if for everyx P A, the closed ball Bpx, εq is contained in A. In particular, the characteristic function of aclosed ball of radius ε in an ultrametric space is ε-locally constant.

The next result says that the Hölder regularity is indeed stronger than the locally constantone.

1This is in particular the case for the sequence spaces of symbolic dynamical systems, see Sections 5.2 and9.2.

2The standard proof using Arzela-Ascoli’s theorem applies with our slightly different definition of the Höldernorms.

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Remark 3.2. Let E and E1 be two metric spaces. If a map f : E Ñ E1 is ε-locally constant,then it is α-Hölder-continuous for every α P s0, 1s. Indeed, for all x, y P E, if dpx, yq ď ε thendpfpxq, fpyqq “ 0 ď c dpx, yqα for all c ą 0. If furthermore E1 “ R and f is bounded, then

supx,yPE, x‰y

|fpxq ´ fpyq|

dpx, yqα“ sup

x,yPE, dpx, yqąε

|fpxq ´ fpyq|

dpx, yqαď

2

εαf8 .

For all ε P s0, 1s and β ą 0, we denote by C ε lc, βb pEq the vector space3 of ε-locally constant

functions f : E Ñ R endowed with the ε lc-norm of exponent β defined by

fε lc, β “ ε´β f8 .

The above remark proves that if β P s0, 1s, the inclusion map from C ε lc, βb pEq into C β

b pEq iscontinuous. We will only use the ε lc-norms in Section 15.4.

3.2 Potentials

In this text, a potential for Γ is a continuous Γ-invariant function rF : T 1X Ñ R. Thequotient function F : ΓzT 1X Ñ R of rF is called a potential on ΓzT 1X. The function rFdefines a continuous Γ-invariant function from GX to R, also denoted by rF , by rF p`q “ rF pv`q.

For all x, y P X, and any geodesic line ` P GX such that `p0q “ x and `pdpx, yqq “ y, letż y

x

rF “

ż dpx,yq

0

rF pvgt`q dt .

Note that for all t P s0, dpx, yqr, the germ vgt` is independent on the choice of such a line `,hence

şyxrF does not depend on the extension ` of the geodesic segment rx, ys. The following

properties are easy to check using the Γ-invariance of rF and the basic properties of integrals:For all γ P Γ

ż γy

γx

rF “

ż y

x

rF ,

for the antipodal map ιż x

y

rF “

ż y

x

rF ˝ ι , (3.2)

and, for any z P rx, ys,ż y

x

rF “

ż z

x

rF `

ż y

z

rF . (3.3)

The period of a loxodromic isometry γ of X for the potential rF is

PerF pγq “

ż γx

x

rF

for any x in the translation axis of γ. Note that, for all α P Γ and n P N´ t0u, we have

PerF pαγα´1q “ PerF pγq, PerF pγ

nq “ n PerF pγq and PerF pγ´1q “ PerF˝ιpγq . (3.4)

In trees, we have the following Lipschitz-type control on the integrals of the potentialsalong segments.

3Note that a linear combination of ε-locally constant functions is again a ε-locally constant function.

43 19/12/2016

Lemma 3.3. When rF is constant or when X is an R-tree, for all x, x1, y, y1 P X, we have

ˇ

ˇ

ˇ

ż y

x

rF ´

ż y1

x1

rFˇ

ˇ ď dpx, x1q supπ´1prx, x1sq

| rF | ` dpy, y1q supπ´1pry, y1sq

| rF | .

Proof. When rF is constant, the result follows from the triangle inequality.Assume that X is an R-tree. Consider the case x “ x1. Let z P X be such that rx, zs “

rx, ys X rx, y1s. Using Equation (3.3) and the fact that dpy, zq ` dpz, y1q “ dpy, y1q, the claimfollows. The general case follows by combining this case x “ x1 and a similar estimate for thecase y “ y1. l

Some form of uniform Hölder-type control of the potential, analogous to the Lipschitz-type one in the previous lemma, will be crucial throughout the present work. The followingDefinition 3.4 formalises this (weaker) assumption.

Definition 3.4. The triple pX,Γ, rF q satisfies the HC-property (Hölder-type control) if rF hassubexponential growth when X is not an R-tree and if there exists κ1 ě 0 and κ2 P s0, 1s suchthat for all x, y, x1, y1 P X with dpx, x1q, dpy, y1q ď 1, we have

ˇ

ˇ

ˇ

ż y

x

rF ´

ż y1

x1

rFˇ

ˇ

ˇ(HC)

ď`

κ1 ` maxπ´1pBpx, dpx,x1qqq

| rF |˘

dpx, x1qκ2 ``

κ1 ` maxπ´1pBpy, dpy,y1qqq

| rF |˘

dpy, y1qκ2 .

By Equation (3.2), pX,Γ, rF ˝ ιq satisfies the HC-property if and only if pX,Γ, rF q does.By Equation (2.3), for every κ P R, the triple pX,Γ, rF ` κq satisfies the HC-property (up tochanging the constant κ1) if and only if pX,Γ, rF q does.

WhenX is assumed to be a Riemannian manifold with pinched sectional curvature, requir-ing the potentials to be Hölder-continuous as in [PauPS] is sufficient to have the HC-property,as we will see below.

Proposition 3.5. The triple pX,Γ, rF q satisfies the HC-property if one of the following con-ditions is satisfied:• rF is constant,• X is an R-tree,• X is a Riemannian manifold with pinched sectional curvature and rF is Hölder-continuous.

Proof. The first two cases are treated in Lemma 3.3, and we may take for them κ1 “ 0 andκ2 “ 1 in the definition of the HC-property.

The claim for Riemannian manifolds follows from the property of at most linear growthof the Hölder-continuous maps (see Remark 3.1) and from [PauPS, Lem. 3.2], with constantsκ1 ą 0 and κ2 P s0, 1s depending only on the Hölder-continuity constants of rF and on thebounds on the sectional curvature of X. l

Remark 3.6. (1) If X “ ĂM is a Riemannian manifold, then T 1X is naturally identified withthe usual Riemannian unit tangent bundle of X. If the potential rF : T 1

ĂM Ñ R is Hölder-continuous for Sasaki’s Riemannian metric on T 1

ĂM , it is a potential as defined in [Rue1] and[PauPS]. Furthermore, the definition of

şyxrF coincides with the one in these references.

(2) The quotient function F is Hölder-continuous when rF is Hölder-continuous.

44 19/12/2016

Let rF , rF ˚ : T 1X Ñ R be potentials for Γ. We say that rF ˚ is cohomologous to rF (see forinstance [Livš]) if there exists a continuous Γ-invariant function rG : T 1X Ñ R, such that, forevery ` P GX, the map t ÞÑ rGpvgt`q is differentiable and

rF ˚pv`q ´ rF pv`q “d

dt |t“0

rGpvgt`q . (3.5)

A potential rF is said to be reversible if rF` and rF´ are cohomologous.When working with Hölder-continuous potentials, the regularity requirement is for rG to

also be Hölder-continuous. Note that the right-hand side of Equation (3.5) does not dependon the choice of the representative ` of its germ v`. In particular, PerF pγq “ PerF˚pγq for anyloxodromic isometry γ if rF and rF ˚ are cohomologous potentials.

3.3 Poincaré series and critical exponents

Let us fix a potential rF : T 1X Ñ R for Γ, and x, y P X.The critical exponent of pΓ, F q is the element δ “ δΓ, F of the extended real line r´8,`8s

defined by

δ “ lim supnÑ`8

1

nln

ÿ

γPΓ, n´1ădpx,γyqďn

eşγyx

rF .

The Poincaré series of pΓ, F q is the map Q “ QΓ, F, x, y : RÑ r0,`8s defined by

Q : s ÞÑÿ

γPΓ

eşγyx p

rF´sq .

If δ ă `8, we say that pΓ, F q is of divergence type if the series QΓ, F, x, ypδq diverges, and ofconvergence type otherwise.

When F “ 0, the critical exponent δΓ, 0 is the usual critical exponent δΓ P s0,`8s of Γ,the Poincaré series QΓ, 0, x, y is the usual Poincaré series of Γ, and we recover the usual notionof divergence or convergence type of Γ.

The Poincaré series of pΓ, F q and its critical exponent make sense even if Γ is elementary(see for instance Lemma 3.7 (10)). The following result collects some of the basic propertiesof the critical exponent. The proofs from [PauPS, Lem. 3.3] generalise to the current setting,replacing the use of [PauPS, Lem. 3.2] by the HC-property.

Lemma 3.7. Assume that pX,Γ, rF q satisfies the HC-property. Then,

(1) the critical exponent δΓ, F and the divergence or convergence of QΓ, F, x, ypsq are inde-pendent of the points x, y P X; they depend only on the cohomology class of rF ;

(2) QΓ, F˝ι, x, y “ QΓ, F, y, x and δΓ, F˝ι “ δΓ, F ; in particular, pΓ, F q is of divergence type ifand only if pΓ, F ˝ ιq is of divergence type ;

(3) the Poincaré series Qpsq diverges if s ă δΓ, F and converges if s ą δΓ, F ;

(4) δΓ, F`κ “ δΓ, F`κ for any κ P R, and pΓ, F q is of divergence type if and only if pΓ, F`κqis of divergence type ;

(5) if Γ1 is a nonelementary subgroup of Γ, denoting by F 1 : Γ1zT 1X Ñ R the map inducedby rF , then δΓ1, F 1 ď δΓ, F ;

45 19/12/2016

(6) if δΓ ă `8, then δΓ ` infπ´1pC ΛΓq

rF ď δΓ, F ď δΓ ` supπ´1pC ΛΓq

rF ;

(7) δΓ, F ą ´8 ;

(8) the map rF ÞÑ δΓ, F is convex, sub-additive, and 1-Lipschitz for the uniform norm on thevector space of real continuous maps on π´1pC ΛΓq ;4

(9) if Γ2 is a discrete cocompact group of isometries of X such that rF is Γ2-invariant,denoting by F 2 : Γ2zT 1X Ñ R the map induced by rF , then

δΓ, F ď δΓ2, F 2 ;

(10) if Γ is infinite cyclic, generated by a loxodromic isometry γ of X, then pΓ, F q is ofdivergence type and

δΓ, F “ max!PerF pγq

λpγq,PerF˝ιpγq

λpγq

)

. l

Examples 3.8. (1) If δΓ is finite and rF is bounded, then the critical exponent δ is finite byLemma 3.7 (6).(2) If X is a Riemannian manifold with pinched negative curvature or when X has a compactquotient, then δΓ is finite. See for instance [Bou].(3) There are examples of pX,Γq with δΓ “ `8 (and hence δ “ `8 if rF is constant), forinstance when X is the complete ideal hyperbolic triangle complex with 3 ideal triangles alongeach edge, see [GaP], and Γ its isometry group. Hence the finiteness assumption of the criticalexponent is nonempty in general. For the type of results treated in this book, it is howevernatural and essential.

We may replace upper limits by limits in the definition of the critical exponents, as follows.

Theorem 3.9. Assume that pX,Γ, rF q satisfies the HC-property. If c ą 0 is large enough,then

δ “ limnÑ`8

1

nln

ÿ

γPΓ, n´cădpx,γyqďn

eşγyx

rF .

If δ ą 0, then

δ “ limnÑ`8

1

nln

ÿ

γPΓ, dpx,γyqďn

eşγyx

rF .

Proof. The proofs of [PauPS, Theo. 4.2], either using the original arguments of [Rob1] validwhen rF is constant, or the super-multiplicativity arguments of [DaPS], extend, using theHC-property (see Definition 3.4) instead of [PauPS, Lem. 3.2]. l

In what follows, we fix a potential rF for Γ such that pX,Γ, rF q satisfies the HC-property.We define rF` “ rF and rF´ “ rF ˝ ι, we denote by F˘ : ΓzT 1X Ñ R their induced maps, andwe assume that δ “ δΓ, F` “ δΓ, F´ is finite.

4That is, if rF , ĂF˚ : T 1X Ñ R are potentials for Γ satisfying the HC-property, inducing F, F˚ : ΓzT 1X Ñ R,and if δΓ, F , δΓ, F˚ ă `8, then δΓ, tF`p1´tqF˚ ď t δΓ, F ` p1´ tq δΓ, F˚ for every t P r0, 1s,

δΓ, F`F˚ ď δΓ, F ` δΓ, F˚ ,

| δΓ, F˚ ´ δΓ, F | ď supvPπ´1pCΛΓq

| ĂF˚pvq ´ rF pvq | .

46 19/12/2016

3.4 Gibbs cocycles

The (normalised) Gibbs cocycle associated with the group Γ and the potential rF˘ is the mapC˘ “ C˘

Γ,F˘: B8X ˆX ˆX Ñ R defined by

pξ, x, yq ÞÑ C˘ξ px, yq “ limtÑ`8

ż ξt

yp rF˘ ´ δq ´

ż ξt

xp rF˘ ´ δq ,

where t ÞÑ ξt is any geodesic ray with endpoint ξ P B8X.We will prove in Proposition 3.10 below that this map is well defined, that is, the above

limits exist for all pξ, x, yq P B8X ˆ X ˆ X and they are independent of the choice of thegeodesic rays t ÞÑ ξt. If rF˘ “ 0, then C´ “ C` “ δΓβ, where β is the Busemann cocycle. IfX is an R-tree, then

C˘ξ px, yq “

ż p

yp rF˘ ´ δq ´

ż p

xp rF˘ ´ δq , (3.6)

where p P X is the point for which r p, ξr “ rx, ξr X ry, ξr ; in particular, the map ξ ÞÑ C˘ξ px, yqis locally constant on the totally discontinuous space B8X.

The Gibbs cocycles satisfy the following equivariance and cocycle properties: For all ξ PB8X and x, y, z P X, and for every isometry γ of X, we have

C˘γξpγx, γyq “ C˘ξ px, yq and C˘ξ px, zq ` C˘ξ pz, yq “ C˘ξ px, yq . (3.7)

For every ` P GX, for all x and y on the image of the geodesic line `, with `´, x, y, `` in thisorder on `, we have

C´`´px, yq “ C```py, xq “ ´C```px, yq “

ż y

xp rF` ´ δq . (3.8)

Proposition 3.10. Assume that pX,Γ, rF q satisfies the HC-property and that δ ă `8.

(1) The maps C˘ : B8X ˆX ˆX Ñ R are well-defined.

(2) For all x, y P X and ξ P B8X, if dpx, yq ď 1, then

|C˘ξ px, yq | ď pκ1 ` δ ` maxπ´1pBpx, dpx,yqqq

| rF |q dpx, yqκ2 ,

with the constants κ1, κ2 of the HC-property. If X is an R-tree, then for all x, y P Xand ξ P B8X, we have

|C˘ξ px, yq | ď dpx, yq maxπ´1prx, ysq

| rF˘ ´ δ| .

(3) The maps C˘ : B8X ˆX ˆX Ñ R are locally Hölder-continuous (and locally Lipschitzwhen X is an R-tree). In particular, they are continuous.

(4) For all r ą 0, x, y P X and ξ P B8X, if ξ belongs to the shadow OxBpy, rq of the ballBpy, rq seen from x, then with the constants κ1, κ2 of the HC-property,

ˇ

ˇ

ˇC˘ξ px, yq `

ż y

xp rF˘ ´ δq

ˇ

ˇ

ˇď 2pκ1 ` δ ` max

π´1pBpy, rqq| rF |q rκ2 .

If X is an R-tree, thenˇ

ˇ

ˇC˘ξ px, yq `

ż y

xp rF˘ ´ δq

ˇ

ˇ

ˇď 2r max

π´1pBpy, rqq| rF˘ ´ δ| .

47 19/12/2016

Proof. (1) The fact that C˘ξ px, yq is well defined when X is an R-tree follows from Equation(3.6).

When X is not an R-tree, let ρ : t ÞÑ ξt be any geodesic ray with endpoint ξ P B8X, lett ÞÑ xt (respectively t ÞÑ yt) be the geodesic ray from x (respectively y) to ξ. Let tx “ βξpx, ξ0q

and ty “ βξpy, ξ0q, so that the quantity β “ ty ´ tx is equal to βξpy, xq (which is independentof ρ), and for every t big enough, we have βξpξt, xt`txq “ βξpξt, yt`tyq “ 0.

ξξ0

y

tx

xt`tx

yt`ty

x

ξtt

ty

Since X is CATp´1q, if t is big enough, then the distances dpξt, xt`txq and dpξt, yt`tyq areat most one, and converge, in a nonincreasing way, exponentially fast to 0 as t Ñ `8. Fors ě 0, let as “

şysy p

rF˘ ´ δq ´şxs´βx p rF˘ ´ δq (which is independent of ρ). We have, using

Equation (HC),ˇ

ˇ

ˇ

´

ż ξt

yp rF˘ ´ δq ´

ż ξt

xp rF˘ ´ δq

¯

´ at`ty

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

´

ż ξt

yp rF˘ ´ δq ´

ż yt`ty

yp rF˘ ´ δq

¯

`

´

ż xt`tx

xp rF˘ ´ δq ´

ż ξt

xp rF˘ ´ δq

¯ˇ

ˇ

ˇ

ď 2pκ1 ` maxπ´1pBpξt, 1qq

| rF ´ δ| q maxtdpξt, xt`txq, dpξt, yt`tyquκ2 ,

which converges to 0 since rF has subexponential growth by the assumptions of the HC-property. Hence in order to prove Assertion (1), we only have to prove that limsÑ`8 asexists.

For all s ě t ě |β|, we have, by the additivity of the integral along geodesics (see Equation(3.3)) and by using again Equation (HC),

|at ´ as| “ˇ

ˇ

ˇ

ż ys

yt

p rF˘ ´ δq ´

ż xs´β

xt´β

p rF˘ ´ δqˇ

ˇ

ˇ

ď pκ1 ` maxπ´1pBpxt´β , 1qq

| rF ´ δ|q dpyt, xt´βqκ2 ` pκ1 ` max

π´1pBpys, 1qq| rF ´ δ| q dpys, xs`βq

κ2 .

Again by the subexponential growth of rF , the above expression converges to 0 as t Ñ `8

uniformly in s, hence limsÑ`8 as exists by a Cauchy type of argument.

(2) Let pξ, x, yq P B8X ˆ X ˆ X. Assertion (2) of Proposition 3.10 follows from Equation(3.6) when X is an R-tree, since rx, ys “ rx, ps Y rp, ys where p be the closest point to y onrx, ξr. When X is not an R-tree, Assertion (2) follows immediately from the HC-property ofpX,Γ, rF˘ ´ δq.

(3) Let pξ, x, yq, pξ1, x1, y1q P B8X ˆX ˆX. By the cocycle property (3.7), we have

|C˘ξ px, yq ´ C˘ξ1 px

1, y1q| ď |C˘ξ px, yq ´ C˘ξ1 px, yq| ` |C

˘ξ1 px, x

1q| ` |C˘ξ1 py, y1q| . (3.9)

48 19/12/2016

First assume that X is an R-tree. Let K be a compact subset of X, and let

εK “ infx,yPK

e´dpx, x0q´dpx, yq ą 0 .

x

ξ1

ξp q

y

Let p, q be the points in X such that rx, ξr X ry, ξr “ rp, ξr and rx, ξr X rx, ξ1r “ rx, qs. Ifdx0pξ, ξ

1q ď εK , then by the definition of the visual distance and by Equation (2.2), we have

e´dpx, qq “ dxpξ, ξ1q ď e´dpx, yq ď e´dpx, pq .

In particular q P rp, ξr, so that rx, ξ1r X ry, ξ1r “ rp, ξ1r. Thus by Equation (3.6), we have

|C˘ξ px, yq ´ C˘ξ1 px, yq | “ 0 .

Therefore, by Equation (3.9) and by the R-tree case of Assertion (2), if dx0pξ, ξ1q ď εK , if

x, y P K and dpx, x1q, dpy, y1q ď 1, then

|C˘ξ px, yq ´ C˘ξ1 px

1, y1q| ď dpx, x1q maxπ´1prx, x1sq

| rF˘ ´ δ| ` dpy, y1q maxπ´1pry, y1sq

| rF˘ ´ δ| .

Since rF is bounded on compact subsets of T 1X, this proves that C˘ is locally Lipschitz.

Let us now consider the case when X is general. For all distinct ξ, ξ1 P B8X, let t ÞÑ ξtand t ÞÑ ξ1t be the geodesic rays from x0 to ξ and ξ1 respectively. By the end of the proof ofAssertion (1), for every compact subset K of X, there exists a1, a2 ą 0 such that for everyx, y P K, we have for η P tξ, ξ1u,

ˇ

ˇ

ˇC˘η px, yq ´

´

ż ηt

yp rF˘ ´ δq ´

ż ηt

xp rF˘ ´ δq

¯ˇ

ˇ

ˇď a1 e

´a2t .

Let T “ ´12 ln dx0pξ, ξ

1q. If T ě 0, by the properties of CATp´1q-spaces, there exist constantsa3, a4 ą 0 such that dpξ2T , ξ

12T q ď a3 and dpξT , ξ

1T q ď a4 e

´T . Hence by Assertion (2), ifdx0pξ, ξ

1q ď mint 1a2

4, 1u (so that T ě 0 and dpξT , ξ1T q ď 1), we have

|C˘ξ px, yq ´ C˘ξ1 px, yq |

ď

ˇ

ˇ

ˇ

ż ξT

yp rF˘ ´ δq ´

ż ξ1T

yp rF˘ ´ δq

ˇ

ˇ

ˇ`

ˇ

ˇ

ˇ

ż ξT

xp rF˘ ´ δq ´

ż ξ1T

xp rF˘ ´ δq

ˇ

ˇ

ˇ` 2 a1 e

´a2T

ď2 pκ1 ` maxπ´1pBpξT , a4qq

| rF˘ ´ δ|q dpξT , ξ1T qκ2 ` 2 a1 e

´a2T .

By the subexponential growth of rF , there exists a5 ą 0 such that

|C˘ξ px, yq ´ C˘ξ1 px, yq | ď a5 e

´κ22T ` 2 a1 e

´a2T ď pa5 ` 2 a1q dx0pξ, ξ1qmint

κ24,a22u .

We now conclude from Equation (3.9) and Assertion (2) as in the end of the above tree casethat C˘ is locally Hölder-continuous.

49 19/12/2016

(4) Let r ą 0, x, y P X and ξ P B8X be such that ξ P OxBpy, rq. Let p be the closest pointto y on rx, ξr , so that dpp, yq ď r. By Equations (3.8) and (3.7), we have

ˇ

ˇ

ˇC˘ξ px, yq `

ż y

xp rF˘ ´ δq

ˇ

ˇ

ˇ“

ˇ

ˇ

ˇC˘ξ px, yq ´ C

˘ξ px, pq ´

ż p

xp rF˘ ´ δq `

ż y

xp rF˘ ´ δq

ˇ

ˇ

ˇ

ď |C˘ξ pp, yq| `ˇ

ˇ

ˇ

ż p

xp rF˘ ´ δq ´

ż y

xp rF˘ ´ δq

ˇ

ˇ

ˇ. (3.10)

First assume that X is an R-tree. Then by Assertion (2) and by Lemma 3.3, we deducefrom Equation (3.10) that

ˇ

ˇ

ˇC˘ξ px, yq `

ż y

xp rF˘ ´ δq

ˇ

ˇ

ˇď 2 dpy, pq max

π´1pry,psq| rF˘ ´ δ| ď 2r max

π´1pBpy, rqq| rF˘ ´ δ| .

In the general case, the result then follows similarly from Equation (3.10) by using Asser-tion (2) and the HC-property. l

3.5 Systems of conductances on trees and generalised electricalnetworks

Let pX, λq be a locally finite metric tree without terminal vertices, letX “ |X|λ be its geometricrealisation, and let Γ be a nonelementary discrete subgroup of IsompX, λq.

Let rc : EX Ñ R be a Γ-invariant function, called a system of (logarithmic) conductancesfor Γ. We denote by c : ΓzEX Ñ R the function induced by rc : EX Ñ R, which we also calla system of conductances on ΓzX.

Classically, an electric network5 (without sources or reactive elements) is a pair pG, ecq,where G is a graph and c : EG Ñ R a function, such that c is reversible: cpeq “ cpeqfor all e P EG, see for example [NaW], [Zem]. In this text, we do not assume our systemof conductances rc to be reversible. In Chapter 6, we will even sometimes assume that thesystem of conductances is anti-reversible, that is, satisfying cpeq “ ´ cpeq for every e P EX.

Two systems of conductances rc, rc1 : EX Ñ R are said to be cohomologous, if there existsa Γ-invariant map f : V XÑ R such that

rc1 ´ rc “ df ,

where for all e P EX, we have

dfpeq “fptpeqq ´ fpopeqq

λpeq.

Proposition 3.11. Let rc : EX Ñ R be a system of conductances for Γ. There exists apotential rF on T 1X for Γ such that for all x, y P V X, if pe1, . . . , enq is the edge path in Xwithout backtracking such that x “ ope1q and y “ tpenq, then

ż y

x

rF “nÿ

i“1

rcpeiqλpeiq .

5A potential in this work is not the analog of a potential in an electric network, we follow the dynamicalsystems terminology as in for example [PauPS].

50 19/12/2016

Proof. Any germ v P T 1X determines a unique edge ev of the tree X, the first one into whichit enters: if ` is any geodesic line whose class in T 1X is v, the edge ev is the unique edge of Xcontaining πpvq whose terminal vertex is the first vertex of X encountered at a positive timeby `. The function rF : T 1X Ñ R defined by

rF pvq “4rcpevq

λpevqmin

dpπpvq, opevqq, dpπpvq, tpevqq(

(3.11)

is a potential on the R-tree X, with rF pvq “ 0 if πpvq P V X.Let us now compute

şyxrF , for all x, y P X. For every λ ą 0, let ψλ : r0, λs Ñ R be the

continuous map defined by ψλptq “ t2

2 if t P r0, λ2 s and ψλptq “λ2

4 ´pλ´tq2

2 if t P rλ2 , λs. Letpe0, e1, . . . , enq be the edge path in X without backtracking such that x P e0 ´ ttpe0qu andy P en ´ topenqu. An easy computation shows that

ż y

x

rF “

n´1ÿ

i“0

rcpeiqλpeiq `4rcpenq

λpenqψλpenq

`

dpy, openqq˘

´4rcpe0q

λpe0qψλpe0q

`

dpx, ope0qq˘

.

If x and y are vertices, the expression simplifies to the sum of the lengths of the edges weightedby the conductances. l

We denote by rFc the potential defined by Equation (3.11) in the above proof, and byFc : ΓzT 1X Ñ R the induced potential. Note that Fc is bounded if c is bounded. We callrFc and Fc the potentials associated with the system of conductances rc and c. This is by nomeans the unique potential with the property required in Proposition 3.11. The followingresult proves that the choice is unimportant.

Given a potential rF : T 1X Ñ R for Γ, let us define a map rcF : EXÑ R by

rcF : e ÞÑ rcF peq “1

λpeq

ż tpeq

opeq

rF . (3.12)

Note that rcF is Γ-invariant, hence it is a system of conductances for Γ. We denote bycF : ΓzEX Ñ R the function induced by rcF : EX Ñ R. Note that rcF`κ “ rcF ` κ for everyconstant κ P R, that rcF is bounded if rF is bounded, and that cFc “ c by the above proposition.

Proposition 3.12. (1) Every potential for Γ is cohomologous to a potential associated witha system of conductances for Γ.

(2) If two systems of conductances rc1 and rc are cohomologous, then their associated poten-tials rFc1 and rFc are cohomologous.

(3) If X has no vertex of degree 2, if two potentials rF ˚ and rF for Γ are cohomologous, thenthe systems of conductances rcF˚ and rcF for Γ are cohomologous.

Hence if X has no vertex of degree 2, the map rF s ÞÑ rcF s from the set of cohomologyclasses of potentials for Γ to the set of cohomology classes of systems of conductances for Γis bijective, with inverse rcs ÞÑ rFcs.

Proof. (1) Let rF be a potential for Γ, and let rF ˚ “ rFcF be the potential associated withthe system of conductances rcF . For all e P EX and t P s0, λpeqr , let ve, t P T 1X be the germof any geodesic line passing at time 0 through the point of e at distance t from opeq. Let

51 19/12/2016

rG : T 1X Ñ R be the map defined by Gpvq “ 0 if πpvq P V X and such that for all e P EX andt P s0, λpeqr ,

rGpve, tq “

ż t

0p rF ˚pve, sq ´ rF pve, sqq ds .

Sinceştpeqopeq

rF “ λpeq rcF peq by construction and λpeq rcF peq “ştpeqopeq

rF ˚ by Proposition 3.11, the

map rG : T 1X Ñ R is continuous. Let ` be a geodesic line. The map t ÞÑ rGpvgt`q is obviouslydifferentiable at time t “ 0 if πp`q R V X, with derivative rF ˚pv`q´ rF pv`q. Since rF ˚pvq vanishesif πpvq P V X by Equation (3.11), and by continuity of rF at such a point, this is still true ifπp`q P V X. Hence rF ˚ and rF are cohomologous, and this proves the first claim.

(2) Assume that rc1 and rc are cohomologous systems of conductances for Γ, and let f : V XÑ Rbe a Γ-invariant function such that rc1 ´ rc “ df . For all e P EX and t P s0, λpeqr , define

rGpve, tq “ λpeq

ż t

0p rFc1pve, sq ´ rFcpve, sqq ds` fpopeqq ,

which is Γ-invariant, whose limit as t Ñ 0 is fpopeqq (independent of the edge e with givenorigin), and whose limit as tÑ λpeq is

λpeq`

rc1peq ´ rcpeq˘

` fpopeqq “ λpeq dfpeq ` fpopeqq “ fptpeqq

(independent of the edge e with given extremity). As above, this proves that rG is continuous,and that Fc1 and Fc are cohomologous.

(3) In order to prove the third claim, assume that rF ˚ and rF are two cohomologous potentialsfor Γ, and let rG : T 1X Ñ R be as in the definition of cohomologous potentials, see Equation(3.5). By the continuity of rG, for all elements v and v1 in T 1X such that πpvq “ πpv1q P V X,we have rGpvq “ rGpv1q, since (by the assumption on the degrees of vertices) the two edges(possibly equal) into which v and v1 enter can be extended to geodesic lines with a commonnegative subray. Hence for every x P V X, the value fpxq “ rGpvxq for every vx P T 1X suchthat πpvxq “ x does not depend on the choice of vx. The map f : V X Ñ R thus definedis Γ-invariant. With the above notation and by Equation (3.12), we hence have, for everye P EX,

rcF˚peq ´ rcF peq “1

λpeq

ż λpeq

0p rF ˚pve, tq ´ rF pve, tq dt “

1

λpeq

ż λpeq

0

d

dtrGpve, tq dt

“1

λpeq

`

rGpvtpeqq ´ rGpvopeqq˘

“fptpeqq ´ fpopeqq

λpeq“ dfpeq .

Hence rcF˚ and rcF are cohomologous. l

Given a metric tree pX,λq, we define the critical exponent of a Γ-invariant system ofconductances rc : EX Ñ R (or of the induced system of conductances c : ΓzEX Ñ R) as thecritical exponent of pΓ, rFcq where rFc is the potential for Γ associated with rc :

δc “ δΓ, Fc .

By Proposition 3.12 (2) and Lemma 3.7 (1), this does not depend on the choice of a potentialrFc satisfying Proposition 3.11.

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Chapter 4

Patterson-Sullivan andBowen-Margulis measures withpotential on CATp´1q spaces

Let X,x0,Γ be as in the beginning of Section 2.2,1 and let rF be a potential for Γ. From nowon, we assume that the triple pX,Γ, rF q satisfies the HC-property of Definition 3.4 and thatthe critical exponent δ “ δΓ, F˘ is finite.

In this chapter, we discuss geometrically and dynamically relevant measures on the bound-ary at infinity of X and on the space of geodesic lines GX. We extend the theory of Gibbsmeasures from the case of manifolds with pinched negative sectional curvature treated in[PauPS] 2 to CATp´1q spaces with the HC-property.

4.1 Patterson densities

A family pµ˘x qxPX of finite nonzero (positive Borel) measures on B8X, whose support is ΛΓ,is a (normalised) Patterson density for the pair pΓ, rF˘q if

γ˚µ˘x “ µ˘γx (4.1)

for all γ P Γ and x P X, and if the following Radon-Nikodym derivatives exist for all x, y P Xand satisfy for (almost) all ξ P B8X

dµ˘xdµ˘y

pξq “ e´C˘ξ px, yq . (4.2)

In particular, the measures µx are in the same measure class for all x P X, and, by Proposition3.10, they depend continuously on x for the weak-star convergence of measures. Note thata Patterson density for pΓ, F˘q is also a Patterson density for pΓ, F˘ ` sq for every s P R,since the definition involves only the normalised potential rF˘´ δ. If F “ 0, we get the usualnotion of a Patterson-Sullivan density (of dimension δΓ) for the group Γ, see for instance[Pat2, Sul2, Nic, Coo, Bou, Rob2].

1That is, X is a geodesically complete proper CATp´1q space, x0 P X is a basepoint, and Γ is a nonele-mentary discrete group of isometries of X.

2See also the previous works [Led, Ham2, Cou, Moh].

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Proposition 4.1. There exists at least one Patterson density for the pair pΓ, rF˘q.

Proof. The Patterson construction (see [Pat1], [Coo]) modified as in [PauPS, Section 3.6]gives the result. l

We refer to Theorem 4.5 for the uniqueness up to scalar multiple of the Patterson densitywhen pΓ, F˘q is of divergence type and to [DaSU, Coro. 17.1.8] for a characterisation of theuniqueness.

The Patterson densities satisfy the following extension of the classical Sullivan shadowlemma (which gives the claim when rF is constant, see [Rob2]), and its corollaries.

If µ is a Borel positive measure on a metric space pX, dq, the triple pX, d, µq is called ametric measure space. A metric measure space pX, d, µq is doubling if there exists c ě 1 suchthat for all x P X and r ą 0

µpBpx, 2 rqq ď c µpBpx, rqq .

Note that, up to changing c, the number 2 may be replaced by any constant larger than1. See for instance [Hei]) for more details on doubling metric measure spaces. We refer forinstance to [DaSU, Ex. 17.4.12] for examples of non-doubling Patterson(-Sullivan) measures,and to [DaSU, Prop. 17.4.4] for a characterisation of the doubling property of the Pattersonmeasures when Γ is geometrically finite and F “ 0.

A family pX,µi, diqiPI of Borel positive measures µi and distances di on a common set Xis called uniformly doubling if there exists c ě 1 such that for all i P I, x P X and r ą 0

µipBdipx, 2 rqq ď c µipBdipx, rqq .

Lemma 4.2. Let pµ˘x qxPX be a Patterson density for the pair pΓ, F˘q, and let K be a compactsubset of X.

(1) [Mohsen’s shadow lemma] If R is large enough, there exists C ą 0 such that for allγ P Γ and x, y P K,

1

Ceşγyx p

rF˘´δq ď µ˘x`

OxBpγy,Rq˘

ď C eşγyx p

rF˘´δq .

(2) For all x, y P X, there exists c ą 0 such that for every n P Nÿ

γPΓ : n´1ădpx, γyqďn

eşγyx p

rF˘´δq ď c .

(3) For every R ą 0 large enough, there exists C “ CpRq ą 0 such that for all γ P Γ andall x, y P K

µ˘x pOxBpγy, 5Rqq ď C µ˘x pOxBpγy,Rqq .

(4) If Γ is convex-cocompact, then the metric measure space pΛΓ, dx, µ˘x is doubling for

every x inX, and the family of metric measure spaces pΛΓ, dx, µ˘x qxPCΛΓ is uniformly

doubling.

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Proof. For the first assertion, the proof of [PauPS, Lem. 3.10] (see also [Cou, Lem. 4] with themultiplicative rather than additive convention, as well as [Moh]) extends, using Proposition3.10 (2), (4) instead of [PauPS, Lem. 3.4 (i), (ii)]. The second assertion is similar to the one of[PauPS, Lem. 3.11 (i)], and the proof of the last two assertions is similar to the one of [PauPS,Prop. 3.12], using Lemma 2.2 instead of [HeP4, Lem. 2.1]. The uniformity in the last assertionfollows from the compactness of ΓzCΛΓ and the invariance and continuity properties of thePatterson densities. l

4.2 Gibbs measures

We fix from now on two Patterson densities pµ˘x qxPX for the pairs pΓ, F˘q.The Gibbs measure rmF on GX (associated with this ordered pair of Patterson densities)

is the measure rmF on GX given by the density

drmF p`q “ eC´`´

px0, `p0qq`C```px0, `p0qq

dµ´x0p`´q dµ

`x0p``q dt (4.3)

in Hopf’s parametrisation with respect to the basepoint x0. The Gibbs measure rmF is inde-pendent of x0, and it is invariant under the actions of the group Γ and of the geodesic flow.Thus, it defines a measure mF on ΓzGX which is invariant under the quotient geodesic flow,called the Gibbs measure on ΓzGX. The proofs of these claims are analogous to the onesof [PauPS, §3.7]. If F “ 0 and the Patterson densities pµ`x qxPX and pµ´x qxPX coincide, theGibbs measure mF coincides with the Bowen-Margulis measure mBM (associated with thisPatterson density), see for instance [Rob2].

Remark 4.3. The (Borel positive) measure given by the density

dλpξ, ηq “ eC´ξ px0, pq`C

`η px0, pq dµ´x0

pξq dµ`x0pηq (4.4)

on B28X is independent of p P sξ, ηr , locally finite and invariant under the diagonal action

of Γ on B28X. It is a geodesic current for the action of Γ on the Gromov-hyperbolic proper

metric space X in the sense of Ruelle-Sullivan-Bonahon, see for instance [Bon] and referencestherein.

Let us now indicate why the terminology of Gibbs measures is indeed appropriate. Thisexplanation will be the point of Proposition 4.4, but we need to give some definitions first.

For all ` P GX and r ą 0, T, T 1 ě 0, the dynamical (or Bowen) ball around ` is

Bp`;T, T 1, rq “

`1 P GX : suptPr´T 1,T s

dp`ptq, `1ptqq ă r(

.

Bowen balls have the following invariance properties: for all s P r´T 1, T s and γ P Γ,

gsBp`;T, T 1, rq “ Bpgs`;T ´ s, T 1 ` s, rq and γBp`;T, T 1, rq “ Bpγ`;T, T 1, rq .

The following inclusion properties of the dynamical balls are immediate: If r ď s, T ě S,T 1 ě S1, then Bp`;T, T 1, rq is contained in Bp`;S, S1, sq. The dynamical balls are almostindependent on r (see [PauPS, Lem. 3.14]): For all r1 ě r ą 0, there exists Tr, r1 ě 0 such that

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for all ` P GX and T, T 1 ě 0, the dynamical ball Bp`;T ` Tr, r1 , T 1 ` Tr, r1 , r1q is contained inBp`;T, T 1, rq.

For every ` P ΓzGX, let us define Bp`;T, T 1, r1q as the image by the canonical projectionGX Ñ ΓzGX of Bpr`;T, T 1, r1q, for any preimage r` of ` in GX.

A pgtqtPR-invariant measure m1 on ΓzGX satisfies the Gibbs property for the potential Fwith constant cpF q P R if for every compact subset K of ΓzGX, there exist r ą 0 and cK, r ě 1such that for all large enough T, T 1 ě 0, for every ` P ΓzGX with g´T

1

`, gT ` P K, we have

1

cK, rď

m1pBp`;T, T 1, rqq

eşT´T 1 pF pvgt`q´cpF qq dt

ď cK, r .

We refer to [PauPS, Sect. 3.8] for equivalent variations on the definition of the Gibbsproperty. The following result shows that the Gibbs measures indeed satisfy the Gibbs prop-erty on the dynamical balls of the geodesic flow, thereby justifying their names. We refer forinstance to [PauPS, Sect. 3.8] for the explanations of the connection with symbolic dynamicsmentioned in the introduction. See also Proposition 4.12 for a discussion of the case when Xis a simplicial tree – here the correspondence with symbolic dynamics is particularly clear.

Proposition 4.4. Let mF be the Gibbs measure on ΓzGX associated with a pair of Pattersondensities pµ˘x qxPĂM for pΓ, rF˘q. Then mF satisfies the Gibbs property for the potential F , withconstant cpF q “ δ.

Proof. The proof is similar to the one of [PauPS, Prop. 3.16] (in which the key Lemma 3.17uses only CATp´1q arguments), up to replacing the use of [PauPS, Lem. 3.4 (1)] by finitelymany applications of Proposition 3.10 (2). l

The basic ergodic properties of the Gibbs measures are summarised in the following result.The case when rF is constant is due to [Rob2], see also [BuM, §6].

Theorem 4.5 (Hopf-Tsuji-Sullivan-Roblin). The following conditions are equivalent

(i) The pair pΓ, F q is of divergence type.

(ii)´ The conical limit set of Γ has positive measure with respect to µ´x for some (equivalentlyevery) x P X.

(ii)` The conical limit set of Γ has positive measure with respect to µ`x for some (equivalentlyevery) x P X.

(iii) The dynamical system pB28X,Γ, pµ

´x b µ`x q|B2

8Xq is ergodic and conservative for some

(equivalently every) x P X.

(iv) The dynamical system pΓzGX, pgtqtPR,mF q is ergodic and conservative.

If one of the above conditions is satisfied, then

(1) the measures µ˘x have no atoms for any x P X,

(2) the diagonal of B8X ˆ B8X has measure 0 for µ´x b µ`x ,

(3) the Patterson densities pµ˘x qxPX are unique up to a scalar multiple, and

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(4) for all x, y P X, as nÑ `8,

maxγPΓ, n´1ădpx,γyqďn

eşγyx

rF˘ “ opeδ nq .

Proof. The proof of the equivalence claim is similar to the one of [PauPS, Theo. 5.4], usingProposition 3.10 (2), (4) instead of [PauPS, Lem. 3.4], and Lemma 4.2 instead of [PauPS,Lem. 3.10]. The claims (1) to (4) are proved as in [PauPS, Sect. 5.3]. l

The following corollary follows immediately from Poincaré’s recurrence theorem and theHopf-Tsuji-Sullivan-Roblin theorem, see [PauPS, Thm. 5.15] for the argument written for themanifold case.

Corollary 4.6. If mF is finite, then

(1) the pair pΓ, F˘q is of divergence type,

(2) the Patterson densities pµ˘x qxPX are unique up to a multiplicative constant and the Gibbsmeasure mF is uniquely defined up to a multiplicative constant.

(3) the support of mF is the image ΩΓ in ΓzGX of rΩΓ “ t` P GX : `˘ P ΛcΓu, and

(4) the geodesic flow is ergodic for mF . l

As the finiteness of the Gibbs measures will be a standing hypothesis in many of thefollowing results, we now give criteria for Gibbs measures to be finite. Recall (see Section2.2) that the discrete nonelementary group of isometries Γ of X is geometrically finite if everyelement of ΛΓ is either a conical limit point or a bounded parabolic limit point of Γ.

Theorem 4.7. Assume that Γ is a geometrically finite discrete group of isometries of X.

(1) If pΓ, F˘q is of divergence type, then the Gibbs measure mF is finite if and only if forevery bounded parabolic limit point p of Γ, the series

ÿ

αPΓp

dpx, αyq eşαyx p rF˘´δq

converges, where Γp is the stabiliser of p in Γ.

(2) If we have δΓp, F˘pă δ, for every bounded parabolic limit point p of Γ with stabiliser Γp

in Γ and with F˘p : ΓpzX Ñ R the map induced by rF˘, then pΓ, F q is of divergencetype. In particular, mF is finite.

When X is a manifold, this result is due to [DaOP, Theo. B] for the case F “ 0, and to[Cou] and [PauPS, Theo. 8.3, 8.4] for the general case of Hölder-continuous potentials. WhenF “ 0 but on much more general assumptions on X with optimal generality, this result is dueto [DaSU, Theo. 17.1.2].

Proof. The proof is similar to the manifold case in [PauPS], which follows closely the proofof [DaOP]. Note that the convergence or divergence of the above series does not depend onthe choice of the sign ˘.

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Let ParΓ be the set of bounded parabolic limit points of Γ. By [Rob2, Lem. 1.9]3, thereexists a Γ-equivariant family pHpqpPParΓ

of pairwise disjoint closed horoballs, with Hp centredat p, such that the quotient

M0 “ Γz`

C ΛΓ´ď

pPParΓ

Hp

˘

is compact. Using Theorem 4.5, the HC-property and Proposition 3.10 instead of [PauPS,Lem. 3.4], the proofs of [PauPS, Theo. 8.3, 8.4] then extend to our situation. l

Recall that the length spectrum of Γ onX is the subgroup of R generated by the translationlengths in X of the elements of Γ.

Recall that a continuous-time 1-parameter group phtqtPR of homeomorphisms of a topo-logical space Z is topologically mixing if for all nonempty open subsets U, V of Z, there existst0 P R such that for all t ě t0, we have U X htV ‰ H.

We have the following result, due to [Bab1, Thm. 1] in the manifold case, with develop-ments by [Rob2] when rF “ 0, and by [PauPS, Sect. 8] for manifolds with pinched negativecurvature.

Theorem 4.8. If the Gibbs measure is finite, then the following assertions are equivalent :

(1) the geodesic flow of ΓzX is mixing for the Gibbs measure,

(2) the geodesic flow of ΓzX is topologically mixing on its nonwandering set, which is thequotient under Γ of the space of geodesic lines in X both of whose endpoints belong toΛΓ.

(3) the length spectrum of Γ on X is not contained in a discrete subgroup of R. l

In the manifold case, the third assertion of Theorem 4.8 is satisfied, for example, if Γ hasa parabolic element, if ΛΓ is not totally disconnected (hence if ΓzX is compact), or if X is asurface or a (rank-one) symmetric space, see for instance [Dal1, Dal2].

Error terms for the mixing property will be described in Chapter 9. The above resultholds for the continuous time geodesic flow when X is a metric tree. See Proposition 4.15 fora version of this theorem for the discrete time geodesic flow on simplicial trees. At least whenX is an R-tree and Γ is a uniform lattice (so that ΓzX is a finite metric graph), we have astronger result under additional regularity assumptions, see Section 9.2.

We end this Section by an elementary remark on the independence of Gibbs measuresupon replacement of the potential F by a cohomologous one.

Remark 4.9. Let rF ˚ : T 1X Ñ R be a potential for Γ cohomologous to rF and satisfying the(HC)-property. As usual, let rF ˚` “ rF ˚ and rF ˚´ “ rF ˚ ˝ ι, and let F ˚ : ΓzT 1X Ñ R be theinduced map. Let rG : T 1X Ñ R be a continuous Γ-invariant function such that, for every` P GX, the map t ÞÑ rGpvgt`q is differentiable and rF ˚pv`q ´ rF pv`q “

ddt |t“0

rGpvgt`q.For all x P X and ξ P B8X, let `x, ξ be any geodesic line with footpoint `x, ξp0q “ x and

endpoint p`x, ξq` “ ξ, and let `ξ, x be any geodesic line with `ξ, xp0q “ x and origin p`ξ, xq´ “ ξ.Note that the value rGpv`x, ξq is independent of the choice of `x, ξ, by the continuity of rG, andsimilarly for rGpv`ξ, xq. In particular, for all γ P Γ, by the Γ-invariance of rG, we have

rGpv`x, γ´1ξq “ rGpv`γx, ξq and rG ˝ ι pv`x, ξq “

rGpv`ξ, xq . (4.5)

3See also [Pau2] for the case of simplicial trees and [DaSU, Thm. 12.4.5] for a greater generality on X.

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Note that rF ˚´ “ rF ˚ ˝ ι and rF´ “ rF ˝ ι are cohomologous, since if rG˚ “ ´ rG ˝ ι, for every` P GX, we have

rF ˚ ˝ ιpv`q ´ rF ˝ ιpv`q “ rF ˚pvι`q ´ rF pvι`q “d

dt |t“0

rGpvgtι`q

“d

dt |t“0

rGpιvg´t`q “d

dt |t“0

rG˚pvgt`q .

As already seen in Lemma 3.7, the critical exponent δΓ, F˚˘ is equal to the critical exponentδΓ, F˘ , and independent of the choice of ˘, and we denote by δ the common value in thedefinition of the Gibbs cocycle.

Note that if C˚˘ “ C˘Γ, F˚˘

is the Gibbs cocycle associated with pΓ, F ˚˘q, then C˚˘ andC˘ are cohomologous:

C˚`ξ px, yq ´ C`ξ px, yq “rGpv`y, ξq ´

rGpv`x, ξq , (4.6)

and similarlyC˚´ξ px, yq ´ C´ξ px, yq “

rG˚pv`y, ξq ´rG˚pv`x, ξq . (4.7)

Let pµ˘x qxPĂM be a Patterson density for pΓ, F˘q. In order to simplify the notation, letrG` “ rG and rG´ “ rG˚. The family of measures pµ˚˘x qxPX defined by setting, for all x P Xand ξ P B8X,

dµ˚˘x pξq “ erG˘pv`x, ξ q dµ˘x pξq , (4.8)

is also a Patterson density for pΓ, F ˚˘q. Indeed, the equivariance property (4.1) for pµ˚˘x qxPXfollows from the one for pµ˘x qxPX and from Equation (4.5). The absolutely continuous property(4.2) for pµ˚˘x qxPX follows from the one for pµ˘x qxPX and Equations (4.6) and (4.7).

Assume that the Patterson density for pΓ, F ˚˘q defined by Equation (4.8) is chosen inorder to construct the Gibbs measure rmF˚ for pΓ, F ˚q on GX. Then using‚ Hopf’s parametrisation with respect to the base point x0 and Equation (4.3) with F

replaced by F ˚ for the first equality,‚ Equations (4.6), (4.7), (4.8) and cancellations for the second equality,‚ the definition of rG˚ “ ´ rG ˝ ι and again Equation (4.3) for the third equality,‚ Equation (4.5) so that ι v``p0q, `´ “ v``´, `p0q , and the fact that we may choose ``´, `p0q “ `

and ``p0q, `` “ ` for the last equality,we have

drmF˚p`q “ eC˚´`´

px0, `p0qq`C˚```px0, `p0qq

dµ˚´x0p`´q dµ

˚`x0p``q dt

“ eC´`´

px0, `p0qq` rG˚pv``p0q, `´q`C```

px0, `p0qq` rGpv``p0q, `´qdµ´x0

p`´q dµ`x0p``q dt

“ e´ rG˝ιpv``p0q, `´

q` rGpv``p0q, `´qdrmF p`q

“ e´rGpv`q` rGpv`q drmF p`q ,

hence rmF˚ “ rmF .In particular, since the Gibbs measure, when finite, is independent up to a multiplicative

constant on the choice of the Patterson densities by Corollary 4.6, we have that mF is finiteif and only if mF˚ is finite, and then

mF˚

mF˚“

mF

mF . (4.9)

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4.3 Patterson densities for simplicial trees

In this Section and the following one, we specialise and modify the general framework of theprevious sections to treat simplicial trees. Recall4 that a simplicial tree X is a metric treewhose edge length map is constant equal to 1. The time 1 map of the geodesic flow pgtqtPRon the space

p

GX of all generalised geodesic lines of the geometric realisation X “ |X|1 of Xpreserves for instance its subset of generalised geodesic lines whose footpoints are at distanceat most 14 from vertices. Since both this subset and its complement have nonempty interiorin

p

GX, the geodesic flow onp

GX has no mixing or ergodic measure with full support. This iswhy we considered the discrete time geodesic flow pgtqtPZ on

p

GX in Section 2.7.

Let X be a locally finite simplicial tree without terminal vertices, and let X “ |X|1be its geometric realisation. Let Γ be a nonelementary discrete subgroup of AutpXq. LetrF : T 1X Ñ R be a potential for Γ, and let rF` “ rF , rF´ “ rF ˝ ι. Let C˘ : B8XˆXˆX Ñ Rbe the associated Gibbs cocycles. Let pµ˘x qxPX be two Patterson densities on B8X for thepairs pΓ, F˘q

Note that only the restrictions of the cocycles C˘ to B8X ˆ V X ˆ V X are useful andthat it is often convenient and always sufficient to replace the cocycles by finite sums involv-ing a system of conductances (as defined in Section 3.5), see below. Furthermore, only therestriction pµ˘x qxPV X of the family of Patterson densities to the set of vertices of X is useful.

Example 4.10. Let X be a simplicial tree and let rc : EXÑ R be a system of conductanceson X. For all x, y in V X and ξ P B8X, with the usual convention on the empty sums, let

c`ξ px, yq “mÿ

i“1

rcpeiq ´nÿ

j“1

rcpfjq

and

c´ξ px, yq “mÿ

i“1

rcpeiq ´nÿ

j“1

rcpfjq ,

where, if p P V X is such that rp, ξr “ rx, ξr X ry, ξr , then pe1, e2, . . . , emq is the geodesic edgepath in X from x “ ope1q to p “ tpemq and pf1, f2, . . . , fnq is the geodesic edge path in X fromv “ opf1q to p “ tpfnq.

f1

fnξ

px e1

em

y f2

With δc defined in the end of Section 3.5 and with C˘ the Gibbs cocycles for pΓ, rFcq, byProposition 3.11, we have, for all ξ P B8X and x, y P V X,

C˘ξ px, yq “ ´ c˘ξ px, yq ` δc dpx, yq ,

and Equation (4.2) givesdµ˘x pξq “ ec

˘ξ px,yq´δc dpx,yq dµ˘y pξq .

4See Section 2.7.

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Using the particular structure of trees, we can prove a version of the Shadow Lemma 4.2where one can take the radius R to be 0. When F “ 0, this result is due to Coornaert [Coo].

Lemma 4.11 (Mohsen’s shadow lemma for trees). Let K be a finite subset of V X. Thereexists C ą 0 such that for all γ P Γ and x, y P K with y P C ΛΓ, we have

1

Ceşγyx p

rF˘´δq ď µ˘x`

Oxtγyu˘

ď C eşγyx p

rF˘´δq .

Proof. The structure of the proof is the same as in Lemma 4.2 with some differences in thedetails towards the end of the argument. Note that C˘ξ px, γyq`

şγyx p

rF˘´δq “ 0 if ξ P Oxtγyu(that is, if γy P rx, ξr ), by Equation (3.8).

First, one argues as in the proof of [PauPS, Lem. 3.10], that it suffices to prove that thereexists C ą 0 such that for all γ P Γ and x, y P K with γy P C ΛΓ, we have

1

Cď µ˘γy

`

Oxtγyu˘

ď C .

Now, the argument for proving the upper bound is the same as in loc. cit., using Proposition3.10 (2) instead of [PauPS, Lem. 3.4 (i)].

In order to prove the lower bound, we assume by contradiction that there exist sequencespxiqiPN, pyiqiPN in K with yi P C ΛΓ and pγiqiPN in Γ such that µ˘γiyipOxitγiyiuq converges to0 as i Ñ `8. Up to extracting a subsequence, since K is finite, we may assume that thesequences pxiqiPN and pyiqiPN are constant, say with value x and y respectively. Since y P C ΛΓ,as every point in C ΛΓ belongs to at least one geodesic line between two limit points of Γ, thegeodesic segment from x to γiy may be extended to a geodesic ray from x to a limit point.Since the support of µ˘z is equal to ΛΓ for any z P X, we have µ˘γiypOxtγiyuq ą 0 for all i P N.Thus, up to taking a subsequence, we can assume that γ´1

i x converges to ξ P ΛΓ (otherwiseby discreteness, we may extract a subsequence so that pγiqiPN is constant, and µ˘γiypOxtγiyuqcannot converge to 0).

Since X is a tree, there exists a positive integer N such that Oγ´1i xtyu “ Oγ´1

N xtyu “ Oξtyu

for all i ě N . As above, Oξtyu meets ΛΓ since y P C ΛΓ, thus µ˘y pOξtyuq ą 0. But for everyi ě N ,

µ˘y`

Oξtyu˘

“ pγ´1i q˚µ

˘γiy

`

Oγ´1i xtyu

˘

“ µ˘γiy`

Oxtγiyu˘

tends to 0 as iÑ `8, a contradiction. l

Let rφ˘µ : V XÑ r0,`8r be the total mass functions of the Patterson densities :

rφ˘µ pxq “ µ˘x

for every x P V X. These maps are Γ-invariant by Equation (4.1), hence they induce mapsφ˘µ : ΓzV X Ñ r0,`8r. In the case of real hyperbolic manifolds and vanishing potentials,the total mass functions have important links to the spectrum of the hyperbolic Laplacian(see [Sul1]). See also [CoP3, CoP5] for the case of simplicial trees and the discrete Laplacian,Section 6.1 for a generalisation of the result of Coornaert and Papadopoulos, and for instance[BerK] for developments in the field of quantum graphs.

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4.4 Gibbs measures for metric and simplicial trees

Let pX, λq be a locally finite metric tree without terminal vertices, and let X “ |X|λ beits geometric realisation. Let Γ be a nonelementary discrete subgroup of AutpX, λq. LetrF : T 1X Ñ R be a potential for Γ. Let pµ˘x qxPV X be two Patterson densities on B8X for thepairs pΓ, F˘q.

The Gibbs measure rmF on the space of discrete geodesic lines GX of X, invariant underΓ and under the discrete time geodesic flow pgtqtPZ of

p

GX, is defined analogously with thecontinuous time case, using the discrete Hopf parametrisation for any basepoint x0 P V X, by

drmF p`q “ eC´`´

px0, `p0qq`C```px0, `p0qq

dµ´x0p`´q dµ

`x0p``q dt , (4.10)

where now dt is the counting measure on Z. We again denote by mF the measure it induceson ΓzGX.

In this Section, we prove that the Gibbs measures in the case of trees satisfy a Gibbsproperty even closer to the one in symbolic dynamics, we give an analytic finiteness criterionof the Gibbs measure for metric trees, and recall the ergodic properties of tree lattices.

As recalled in the introduction, Gibbs measures were first introduced in statistical me-chanics and consequently in symbolic dynamics, see for example [Bowe2], [ParP], [PauPS]. Inorder to motivate the terminology used here, we recall the definition of a Gibbs measure forthe full two-sided shift on a finite alphabet:5 Let Σn “ t1, 2, . . . , nu

Z be the product space ofsequences x “ pxnqnPZ indexed by Z in the finite set t1, 2, . . . , nu, and let σ : Σn Ñ Σn be theshift map defined by σppxnqnPZq “ pxn`1qnPZ. A shift-invariant probability measure µ on Σn

satisfies the Gibbs property for an energy function φ : Σn Ñ R if

1

Cďµpra´m´ , a´m´`1, . . . , am`´1, am`sq

e´P pm´`m``1q`

řm`k“´m´

φpσkxqď C

for some constants C ě 1 and P P R (called the pressure) and for all m˘ in N and x in thecylinder ra´m´ , a´m´`1, . . . , am`´1, am`s that consists of those x P Σn for which xk “ ak forall k P r´m´,m`s.

Let x´, x` P V X and let x0 P V XX rx´, x`s. Let us define the tree cylinder of the triplepx´, x0, x`q by

rx´, x0, x`s “ t` P GX : `˘ P Ox0tx˘u, `p0q “ x0u .

These cylinders are close to dynamical balls that have been introduced in Section 4.2, and theparallel with the symbolic case is obvious, as this cylinder is the set of geodesic lines whichcoincides on r´m´,m`s, where m˘ “ dpx0, x˘q, with a given geodesic line passing throughx˘ and through x0 at time t “ 0. The Gibbs measure rmF on the space of discrete geodesiclines GX satisfies a variant of the Gibbs property which is even closer to the one in symbolicdynamics than the general case described in Proposition 4.4.

Proposition 4.12 (Gibbs property). Let K be a finite subset of V X X C ΛΓ. There existsC 1 ą 1 such that for all γ P Γ and x˘ P ΓK and all x0 P V XX rx´, x`s,

1

C 1ď

rmF prx´, x0, x`sq

e´δ dpx´, x`q`

şx`x´

rFď C 1

5See Section 5.1 for the appropriate definition when the alphabet is countable.

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Proof. The result is immediate if dpx´, x`q is bounded, since the above denominator andnumerator take only finitely many values, and the numerator is nonzero since x˘ P C ΛΓ,hence rx´, x0, x`s meets the support of rmF . We may hence assume that dpx´, x`q ě 2.Using the invariance of rmF under the discrete time geodesic flow, we may thus assume thatx0 ‰ x´, x`.

Using the discrete Hopf parametrisation with respect to the vertex x0, we have, by Lemma4.11,

rmF prx´, x0, x`sq “ µ´x0pOx0tx´uqµ

`x0pOx0tx`uq

ď C2 eşx´x0p rF˝ι´δq

eşx`x0p rF´δq

“ C2eşx`x´p rF´δq

.

This proves the upper bound in Proposition 4.12 with C 1 “ C2 and the lower bound followssimilarly. l

Next, we give a finiteness criterion of the Gibbs measure for metric trees in terms of thetotal mass functions of the Patterson densities, extending the case when Γ is torsion free andrF “ 0, due to [CoP4, Theo. 1.1].

Proposition 4.13. Let pX, λ,Γ, rF q be as in the beginning of this Section.

(1) If pX, λq is simplicial and ¨ 2 is the Hilbert norm of L2pΓzV X, volΓzzXq, we have

mF ď φ`µ 2

φ´µ 2 .

(2) In general, with ¨ 2 the Hilbert norm of L2pΓzEX,TvolΓzzX,λq, we have6

mF ď φ`µ ˝ o2 φ

´µ ˝ o2 .

Proof. (1) The simplicial assumption on pX, λq means that all edges have length 1. Thespace ΓzGX is the disjoint union of the subsets t` P ΓzGX : πp`q “ `p0q “ rxsu as the classrxs “ Γx of x P V X ranges over ΓzV X. By Equation (4.10), using Hopf’s decomposition withrespect to the basepoint x, we have

dprmF q|t`PGX : `p0q“xup`q “ dµ´x p`´q dµ`x p``q .

Let ∆rxs be the unit Dirac mass at rxs. By ramified covering arguments, we hence have thefollowing equality of measures on the discrete set ΓzV X:

π˚mF “ÿ

rxsPΓzV X

1

|Γx|µ´x ˆ µ

`x

`

tp`´, ``q P B28X : x P s`´, ``ru

˘

∆rxs . (4.11)

Thus, using the Cauchy-Schwarz inequality,

mF “ π˚mF ďÿ

rxsPΓzV X

1

|Γx|µ´x ˆ µ

`x “ xφ

´µ , φ

`µ y2

ď φ`µ 2φ´µ 2

.

This proves Assertion (1) of Proposition 4.13.6Recall that o : ΓzEXÑ ΓzV X is the initial vertex map, see Section 2.7.

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(2) The argument is similar to the proof of the simplicial case. Since the singletons in R havezero Lebesgue measure, the space ΓzGX is, up to a measure zero subset for mF , the disjointunion for res P ΓzEX of the sets Ares consisting of the elements ` P ΓzGX such that `p0qbelongs to the interior of the edge res. We fix a representative e of each res P ΓzEX. For everyt P r0, λpeqs, let et be the point of e at distance t from opeq. By Equation (4.3), using Hopf’sdecomposition with respect to the basepoint opeq in Ares, we have as above

mF “ÿ

resPΓzEX

1

|Γe|

ż

`´PBeX

ż

``PBeX

ż λpeq

0eC´`´

popeq, etq`C```popeq, etq

dµ´opeqp`´q dµ`

opeqp``q dt .

Since `´, opeq, et, `` are in this order on the geodesic line ` with `´ P BeX and `` P BeX, wehave C´`´popeq, etq ` C

```popeq, etq “ 0 by Equation (3.8). Hence

mF “ÿ

resPΓzEX

λpeq

|Γe|µ´opeqpBeXq µ

`

opeqpBeXq

ďÿ

resPΓzEX

λpeq

|Γe|µ´opeq µ

`

opeq “ xφ´µ ˝ o, φ

`µ ˝ oy2 ď φ

´µ ˝ o2 φ

`µ ˝ o2 ,

which finishes the proof. l

Let us give some corollaries of this proposition in the case of simplicial trees. It followsfrom Assertion (1) of Proposition 4.13 that if the L2-norms of the total mass of the Pattersondensities are finite, then the Gibbs measure mF is finite. Taking rF “ 0 and pµ`x qxPV X “

pµ´x qxPV X, so that the Gibbs measure mF is the Bowen-Margulis measure mBM, it followsfrom this proposition that

mBM ď φ˘µ 2

2ď VolpΓzzXq sup

xPV Xµ˘x

2 . (4.12)

In particular, if Γ is a lattice in X and if the total mass of the Patterson density is bounded,then the Bowen-Margulis measure mBM is finite.

The following statement summarises the basic ergodic properties of the lattices of X whenF “ 0.

Proposition 4.14. Let pX, λq be a metric or simplicial tree, with geometric realisation X.Assume that pX, λq is uniform and that Γ is a lattice in AutpX, λq. Then

(1) Γ is of divergence type, and its critical exponent δΓ is the Hausdorff dimension of anyvisual distance dx on B8X “ ΛΓ;

(2) the Patterson density pµxqxPX coincides, up to a scalar multiple, with the family ofHausdorff measures pµHaus

x qxPX of dimension δΓ of the visual distances pB8X, dxq; inparticular, it is AutpX, λq-equivariant: for all γ P AutpX, λq, we have γ˚µx “ µγx;

(3) the Bowen-Margulis measure rmBM of Γ on GX is AutpX, λq-invariant, and the Bowen-Margulis measure mBM of Γ on ΓzGX is finite.

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Proof. Let Γ1 be any uniform lattice of pX, λq, which exists since the metric tree pX, λq isuniform. It is well-known (see for instance [Bou]) that the critical exponent δΓ1 of Γ1 is finiteand equal to the Hausdorff dimension of any visual distance pB8X, dxq, and that the familypµHausx qxPV X of Hausdorff measures of the visual distances pB8X, dxq is a Patterson density

for any discrete nonelementary subgroup of AutpX, λq with critical exponent equal to δΓ1 .By [BuM, Coro. 6.5(2)], the lattice Γ in AutpX, λq is of divergence type and δΓ “ δΓ1 . By

the uniqueness property of the Patterson densities when Γ is of divergence type (see Theorem4.5), the family pµxqxPV X coincides, up to a scalar multiple, with pµHaus

x qxPV X.As the graph Γ1zX is compact, the total mass function of the Hausdorff measures of the

visual distances is bounded, hence so is pµxqxPV X. By Proposition 4.13, since Γ is a latticeof X, this implies that the Bowen-Margulis measure mBM of Γ is finite. l

Note that as in [DaOP], when pX, λq (or its minimal nonempty Γ-invariant subtree) is notassumed to be uniform, there are examples of Γ that are lattices (or are geometrically finite)whose Bowen-Margulis measure mBM is infinite, see Section 15.5 for more details.

Let us now discuss the mixing properties of the discrete time geodesic flow on ΓzGX forthe Gibbs measure mF .

Let LΓ be the length spectrum of Γ, which is, in this simplicial case, the subgroup of Zgenerated by the translation lengths of the elements of Γ.

Since dpx, γxq “ 2 dpx,Axγq ` `pγq if an isometry γ of X is loxodromic and dpx, γxq “2 dpx,Fixγq if γ is elliptic with fixed point set Fixγ , the following assertions are equivalent :

p1q LΓ Ă 2Zp2q @ x P X, @ γ P Γ, dpx, γxq P 2Z . (4.13)

Let GevenX (respectivelyp

Geven X) be the subset of GX (respectivelyp

GX) that consists ofthe geodesic lines (respectively generalised geodesic lines) in X whose origin is at an evendistance from the basepoint x0.

Recall that a discrete-time 1-parameter group phnqnPZ of homeomorphisms of a topologicalspace Z is topologically mixing if for all nonempty open subsets U, V of Z, there exists n0 P Nsuch that for all n ě n0, we have U X hnV ‰ H.

The following result is well-known. When rF “ 0, see for instance [Rob2] for the equivalenceof the first, second and fourth claims, and the arguments of [BrP2, Prop. 3.3] for what remainsto be proved.

Proposition 4.15. Assume that the smallest nonempty Γ-invariant simplicial subtree of Xis uniform, without vertices of degree 2, and that mF is finite. Then the following assertionsare equivalent:• the length spectrum of Γ is nonarithmetic, that is LΓ “ Z;• the discrete time geodesic flow on ΓzGX is topologically mixing on its nonwandering set;• the quotient graph ΓzX is not bipartite;• the Gibbs measure mF is mixing under the discrete time geodesic flow pgtqtPZ on ΓzGX.Otherwise LΓ “ 2Z, and the square of the discrete time geodesic flow pg2tqtPZ is topologicallymixing on the nonwandering subset of ΓzGevenX and mixing for the restriction of the Gibbsmeasure mF to ΓzGevenX. l

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By Proposition 4.14, the general assumptions of Proposition 4.15 are satisfied if X isuniform, without vertices of degree 2, Γ is a lattice of X and rF “ 0. Thus, if we assumefurthermore that ΓzX is not bipartite, then the Bowen-Margulis measure mBM of Γ is mixingunder the discrete time geodesic flow on ΓzGX.

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Chapter 5

Symbolic dynamics of geodesic flowson trees

5.1 Two-sided topological Markov shifts

In this short and independent Section, that will be used in Sections 5.2, 5.3, 5.4, 9.2 and 9.3,we recall some definitions concerning symbolic dynamics on countable alphabets.1

A (two-sided, topological) Markov shift2 is a topological dynamical system pΣ, σq con-structed from a countable discrete alphabet A and a transition matrix A “ pAi, jqi, jPA P

t0, 1uAˆA , where Σ is the closed subset of the topological product space A Z defined by

Σ “

x “ pxnqnPZ P A Z : @ n P Z, Axn,xn`1 “ 1u ,

and σ : Σ Ñ Σ is the (two-sided) shift defined by

pσpxqqn “ xn`1

for all x P Σ and n P Z. Note that to be given pA , Aq is equivalent to be given an orientedgraph with countable set of vertices A (and set of oriented edges a subset of A ˆ A ) andwith incidence matrix A such that Ai, j “ 1 if there is an oriented edge from the vertex i tothe vertex j and Ai, j “ 0 otherwise.

For all p ď q P Z, a finite sequence panqpďnďq P A tp,...,qu is admissible (or A-admissiblewhen we need to make A precise) if Aan, an`1 “ 1 for all n P tp, . . . , q ´ 1u. A topologicalMarkov shift is transitive if for all x, y P A , there exists an admissible finite sequence panqpďnďqwith ap “ x and aq “ y. This is equivalent to require the dynamical system pΣ, σq to betopologically transitive: for all nonempty open subsets U, V in Σ, there exists n P Z such thatAX σnpBq ‰ H.

Note that the product space A Z is not locally compact when A is infinite. When thematrix A has only finitely many nonzero entries on each line and each colum, then pΣ, σq isalso called a subshift of finite type (on a countable alphabet). The topological space Σ is then

1See for instance [Kit, Sar2].2Note that the terminology could be misleading, a topological Markov shift comes a prori without a measure,

and many probability measures invariant under the shift do not satisfy the Markov chain property that theprobability to pass from one state to another depends only on the previous state, not of all past states.

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locally compact: By diagonal extraction, for all p ď q in Z and ap, ap`1, . . . , aq´1, aq in A ,every cylinder

rap, ap`1, . . . , aq´1, aqs “

pxnqnPZ P Σ : @ n P tp, . . . , qu, xn “ an(

is a compact open subset of Σ.Given a continuous map Fsymb : Σ Ñ R and a constant cFsymb

P R, we say that a measureP on Σ, invariant under the shift σ, satisfies the Gibbs property3 with Gibbs constant cFsymb

for the potential Fsymb if for every finite subset E of the alphabet A , there exists CE ě 1such that for all p ď q in Z and for every x “ pxnqnPZ P Σ such that xp, xq P E, we have

1

C Eď

Pprxp, xp`1, . . . , xq´1, xqsq

e´cFsymb

pq´p`1q`řqn“p Fsymbpσnxq

ď CE . (5.1)

5.2 Coding discrete time geodesic flows on simplicial trees

Let X be a locally finite simplicial tree without terminal vertices, with X “ |X|1 its geometricrealisation. Let Γ be a nonelementary discrete subgroup of AutX, and let rF : T 1X Ñ R bea potential for Γ.

In this Section, we give a coding of the discrete time geodesic flow pgtqtPZ on the nonwan-dering subset of ΓzGX by a locally compact transitive (two-sided) topological Markov shift.This explicit construction will be useful later on to study the variational principle (see Section5.4) and rates of mixing (see Section 9.2).

The main technical aspect of this construction, building on [BrP2, §6], is to allow the casewhen Γ has torsion. When Γ is torsion free and ΓX is finite, the construction is well-known,we refer for instance to [CoP6] for a more general setting when the potential is 0. In orderto consider for instance non-uniform tree lattices, it is important to allow torsion in Γ. Ourdirect approach also avoids the assumption that the discrete subgroup Γ is full, that is, equalto the subgroup consisting of the elements g P AutpXq such that p ˝ g “ p where p : XÑ ΓzXis the canonical projection, as in [Kwo] (building on [BuM, 7.3]).

Let X1 be the minimal nonempty Γ-invariant simplicial subtree of X, whose geometricrealisation is C ΛΓ. Since we are only interested in the support of the Gibbs measures, wewill only code the geodesic flow on the non-wandering subset ΓzX1 of ΓzGX. The sameconstruction works with the full space ΓzGX, but the resulting Markov shift is then notnecessarily transitive.

Let pY, G˚q “ ΓzzX1 be the quotient graph of groups of X1 by Γ (see for instance Example2.10), and let p : X1 Ñ Y “ ΓzX1 be the canonical projection. We denote by r1s “ H thetrivial double coset in any double coset set HzGH of a group G by a subgroup H.

We consider the alphabet A consisting of the triples pe´, h, e`q where‚ e˘ P EY satisfy tpe´q “ ope`q and‚ h P ρe´pGe´qzGope`qρ e`pGe`q satisfy h ‰ r1s if e

` “ e´.

This set is countable (and finite if and only if the quotient graph ΓzX1 is finite), we endowit with the discrete topology. We consider the (two-sided) topological Markov shift withalphabet A and transition matrix Ape´, h, e`q, pe1´, h1, e1`q “ 1 if e` “ e1´ and 0 otherwise.

3Note that some references have a stronger notion of Gibbs measure (see for instance [Sar1]), with theconstant C independent of E.

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Note that this matrix A “ pAi,jqi,jPA has only finitely many nonzero entries on each line andeach column, since X1 is locally finite and Γ has finite vertex stabilisers in X1. We considerthe subspace

Σ “

pe´i , hi, e`i qiPZ : @ i P Z, e`i´1 “ e´i

(

of the product space A Z, and the shift σ : Σ Ñ Σ defined by pσpxqqi “ xi`1 for all pxiqiPZ inΣ and i P Z. As seen above, Σ is locally compact.

Let us now construct a natural coding map Θ from ΓzGX1 to Σ, by slightly modifying theconstruction of [BrP2, §6].

Ąei`1

geiĆtpeiq

rei

γi`1 p ei

fi fi`1

γi

g ei`1

ei`1

`

in Xin pY, G˚q “ ΓzzX

hi`1p`q P Gtpeiq

For every discrete geodesic line ` P GX1, for every i P Z, let fi “ fip`q be the edge ofX1 whose geometric realisation is `pri, i ` 1sq with origin fpiq and endpoint fpi ` 1q, and letei “ ppfiq, which is an edge in Y. Let us use the notation of Example 2.10: we fix lifts reand rv of every edge e and vertex v of Y in X1 such that re “ re, and elements ge P Γ suchthat ge Ątpeq “ tpreq. Since ppreiq “ ei “ ppfiq, there exists γi “ γip`q P Γ, well defined up tomultiplication on the left by an element of Gei “ Γ

rei , such that γifi “ rei for all i P Z.We define e´i p`q “ ei´1, e`i p`q “ ei, and

hip`q “ g ´1e´i

γi´1p`q γip`q´1 g

e`i. (5.2)

Since for every edge e of Y the structural monomorphism

ρe : Ge “ Γre ÝÑ Gtpeq “ Γ

Ątpeq

is the map g ÞÑ g´1e gge, the double coset of hip`q in ρe´i

pGe´iqzGope`i q

ρe`ipGe`q does not

depend on the choice of the γi’s, and we again denote it by hip`q.

The next result shows that, under the only assumptions on Γ that it is discrete andnonelementary, the time-one discrete geodesic flow g1 on its nonwandering subset of ΓzGX istopologically conjugate to a locally compact transitive (two-sided) topological Markov shift.

Theorem 5.1. If X1 “ C ΛΓ, the map Θ : ΓzGX1 Ñ Σ defined by

Γ` ÞÑ pe´i p`q, hip`q, e`i p`qqiPZ

is a homeomorphism which conjugates the time-one discrete geodesic flow g1 and the shift σ,and the topological Markov shift pΣ, σq is a locally compact and transitive.

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Furthermore, if we endow ΓzGX1 with the quotient distance of4

dp`, `1q “ e´max

nPN : `|r´n,ns “ `1|r´n,ns

(

on GX1 and Σ with the distance5

dpx, x1q “ e´max

nPN : @ i P t´n,...,nu, xi “ x1i

(

,

then Θ is a bilipschitz homeomorphism.Finally, if X1 is a uniform tree without vertices of degree at most 2, if the Gibbs measure

mF of Γ is finite, and if the length spectrum LΓ of Γ is equal to Z, then the topological Markovshift pΣ, σq is topologically mixing.

The following diagram hence commutes

ΓzGX1 g1

ÝÑ ΓzGX1

Θ Ó Ó Θ

ΣσÝÑ Σ .

Note that when Y is finite (or equivalently when Γ is cocompact), the alphabet A is finite(hence pΣ, σq is a standard subshift of finite type). When furthermore the vertex groups ofpY, G˚q are trivial (or equivalently when Γ acts freely, and in particular is a finitely generatedfree group), this result is well-known, but it is new if the vertex groups are not trivial.Compare with the construction of [CoP6], whose techniques might be applied since Γ is word-hyperbolic if Y is finite, up to replacing Gromov’s (continuous time) geodesic flow of Γ by the(discrete time) geodesic flow on GX1, thus avoiding the suspension part (see also the end ofloc. cit. when Γ is a free group).

Proof. For all ` P GX1 and γ P Γ, we can take γipγ`q “ γip`qγ´1, and since ppγfiq “ ppfiq,

we have e˘i pγ`q “ e˘i p`q and hipγ`q “ hip`q, hence the map Θ is well defined.By construction, the map Θ is equivariant for the actions of g1 on ΓzGX1 and σ.For every N P N, if `, `1 P GX1 are close enough, then for ´N ´ 1 ď i ď N ` 1, we have

fip`q “ fip`1q and we may take γip`q “ γip`

1q, so that e˘i p`q “ e˘i p`1q and hip`q “ hip`

1q for´N ď i ď N . Therefore Θ is continuous.

Furthermore, with the distances indicated in the statement of Theorem 5.1, if `, `1 P GX1satisfy `|r´n, ns “ `1

|r´n, ns for some n P N, then we have e˘i p`q “ e˘i p`1q for ´n ď i ď n ´ 1,

and we may take γip`q “ γip`1q for ´n ď i ď n´ 1, so that hip`q “ hip`

1q for ´n ď i ď n´ 1.Therefore, we have

dpΘpΓ`q,ΘpΓ`1q q ď e dpΓ`,Γ`1q ,

and Θ is Lipschitz.

Let us construct an inverse Ψ : Σ Ñ ΓzGX1 of θ, by a more general construction that willbe useful later on. Let I be a nonempty interval of consecutive integers in Z, either finite orequal to Z (the definition of the inverse of θ only requires the second case I “ Z). For alle´, e` P EY such that tpe´q “ ope`q, we fix once and for all a representative of every doublecoset in ρe´pGe´qzGope`qρ e`pGe`q, and we will denote this double coset by its representative.

4 with the convention that maxH “ 0 when we consider the empty set H as a subset of N.5See previous footnote.

70 19/12/2016

Let w “ pe´i , hi, e`i qiPI be a sequence indexed by I in the alphabet A such that for all

i P I such that i ´ 1 P I, we have e`i´1 “ e´i (when I is finite, this simply means that wis an A-admissible sequence in A , and when I “ Z, this simply means that w P Σ). Inparticular, the element hi P Gope`i q “ Γ

Čope`i qis the chosen representative of its double coset

ρe´ipGe´i

q hi ρ e`ipGe`i

q.For every i P I, note that

ophi g´1

e`i

Ăe`i q “ op g ´1

e`i

Ăe`i q “Čope`i q “

Ćtpe´i q “ tp g ´1e´i

Ăe´i q .

But hi g ´1

e`i

Ăe`i is not the opposite edge of the edge g ´1e´i

Ăe´i , since the double coset of hi is

not the trivial one r1s when e`i “ e´i , hence hi does not fix g´1e´i

Ăe´i . Therefore the length 2

edge path (see the picture below)

p g ´1e´i

Ăe´i , hi g´1

e`i

Ăe`i q

is geodesic.

e´i “ e`i´1 e`i`1e`i “ e´i`1

g ´1e´i

Ăe´i

hi

hi`1

Čope`i q

Y

X

fi´1 fi

αi´1

αi

fi`1

p

g ´1

e`i

Ăe`i

g ´1e`i

Ăe`i

g ´1

e`ige`i

Let us construct by induction a geodesic segment rw in X1 (which will be a discrete geodesicline if I “ Z), well defined up to the action of Γ, as follows.

We fix i0 P I (for instance i0 “ 0 if I “ Z or i0 “ min I if I is finite), and αi0 P Γ. Let usdefine

fi0 “ fi0pwq “ αi0 g´1e`i0

Ăe`i0 .

Let us then define

αi0´1 “ αi0´1pwq “ αi0 g´1

e`i0ge`i0

h ´1i0

and fi0´1 “ fi0´1pwqαi0´1 g´1

e´i0

Ăe´i0 .

We have αi0´1 hi0 g´1

e`i0

Ăe`i0 “ fi0 and pfi0´1, fi0q is a geodesic edge path of length 2 (as the

image by αi0´1 of such a path).

71 19/12/2016

Let i ´ 1, i1 P I be such that i1 ď i0 ď i ´ 1. Assume by increasing induction on i anddecreasing induction on i1 that a geodesic edge path pfi1´1 “ fi1´1pwq, . . . , fi´1 “ fi´1pwqq inX1 and a sequence pαi1 “ αi1´1pwq, . . . , αi´1 “ αi´1pwqq in Γ have been constructed such that

fj “ αj g´1

e`j

Ăe`j and αj “ αj´1 hj g´1

e`jge`j

for every j P N such that i1 ´ 1 ď j ď i´ 1, with besides j ě i1 for the equality on the right.If i does not belong to I, we stop the construction on the right hand side at i ´ 1. If on

the contrary i P I, let us define (see the above picture)

αi “ αi´1 hi g´1

e`ige`i

and fi “ fipwq “ αi g´1

e`i

Ăe`i .

Thenpfi´1, fiq “

`

αi´1 g´1

e´i

Ăe´i , αi´1 hi g´1

e`i

Ăe`i˘

,

is a geodesic edge path of length 2 (as the image by αi´1 of such a path). As an edge pathis geodesic if and only if it has no back-and-forth, pfi1 , . . . , fiq is a geodesic edge path in X1.Thus the construction holds at rank i on the right.

If i1´ 1 does not belong to I, we stop the construction on the left side at i1. Otherwise weproceed as for the construction of αi0´1 and fi0´1 in order to construct αi´1 and fi´1 withthe required properties.

If I “ rp, qs XZ with p ď q P Z, let I 1 “ rp´ 1, qs XZ. If I “ Z, let I 1 “ Z. We have thusconstructed a geodesic edge path

pfiqiPI 1 “ pfipwqqiPI 1 (5.3)

in X1. We denote by rw its parametrisation by R if I “ Z and by rp´ 1, q` 1s if I “ rp, qsXZ,in such a way that rwpiq “ opfiq for all i P I. In particular, fi “ rwpri, i ` 1sq for all i P I 1.When I “ rp, qs X Z, we consider rw as a generalised discrete geodesic line, by extending it toa constant on s ´ 8, p´ 1s and on rq,`8r .

The orbit Γ rw of rw does not depend on the choice of αi0 , since replacing αi0 by α1i0 replacesfi by α1i0α

´1i0fi for all i P I 1, hence replaces rw by α1i0α

´1i0

rw. This also implies that Γ rw doesnot depend on the choice of i0 P I.

Assume from now on that I “ Z, and define Ψ : Σ Ñ ΓzGX1 by

Ψpwq “ Γ rw .

With the distances indicated in the statement of Theorem 5.1, let w “ pe´i , hi, e`i qiPZ and

w1 “ pe1i´, h1i, e

1i`qiPI in Σ satisfy e˘i “ e1i

˘ and hi “ h1i for all i P t´n, . . . , nu for somen P N. Then we may take the same i0 “ 0 and αi0 in the construction of rw and Ăw1. We thushave αipwq “ αipw

1q and fipwq “ fipw1q for ´n ď i ď n. Therefore, we have

dpΨpwq,Ψpw1q q ď dpw,w1q ,

and Ψ is Lipschitz.Let us prove that Ψ is indeed the inverse of Θ. As in the construction of Θ, for all ` P GX1

and i P Z, we define fi “ `pri, i ` 1sq, e`i “ ppfiq and e´i “ e`i´1. We denote by γ1i P Γ an

element sending fi to g ´1e´i

Ăe´i for all i P Z (see the picture below): with the notation above

the statement of Theorem 5.1, we have γ1i “ g ´1e´i

γip`q.

72 19/12/2016

e´i “ e`i´1 e`i`1e`i “ e´i`1Y

fi´1 fi fi`1

g ´1e`i

Ăe`i

g ´1e´i

Ăe´i

h1i

h1i`1

γ1i

X

γ1i´1

Čope`i q

γ1i´1fi

p

g ´1

e`i

Ăe`i

g ´1

e`ige`i

Then γ1i is well defined up to multiplication on the left by an element of Γg ´1

e´i

Ăe´i“ ρe´i

pGe´iq.

Let h1i be an element in Gope`i q sending g´1

e`i

Ăe`i to γ1i´1fi. It exists since these two edges have

the same origin Čope`i q, and same image by p:

ppγ1i´1fiq “ ppfiq “ e`i “ ppĂe`i q “ pp g ´1

e`i

Ăe`i q .

Furthermore, it is well defined up to multiplication on the right by an element of Γg ´1

e`i

Ăe`i“

ρe`ipGe`i

q, and we have (see the above picture)

γ1i´1 γ1i´1

g ´1e´i

ge`iP h1i ρe`i

pGe`iq

By the construction of Θ (see Equation (5.2) with γ1i “ g ´1e´i

γip`q for all i P Z), we have

Θp`q “ pe´i , ρe´ipGe´i

q h1i ρe`ipGe`i

q, e`i p`qqiPZ .

Let hi be the chosen representative of the double coset ρe´i pGe´i q h1i ρe`i

pGe`iq : there exists

α P ρe´ipGe´i

q and β P ρe`ipGe`i

q such that hi “ αh1iβ. Up to replacing γ1i by α´1γ1i and h

1i by

h1iβ, we then may have h1i “ hi. By taking αi0 “ γ1i0´1, we have αi “ γ1i

´1 for all i P Z, andan inspection of the above two constructions gives that Θ ˝Ψ “ id and Ψ ˝Θ “ id.

Since the discrete time geodesic flow is topologically transitive on its nonwandering subsetand by conjugation, the topological Markov shift pΣ, σq is topologically transitive.

If X1 is a uniform tree without vertices of degree at most 2, and if the length spectrum ofΓ is nonarithmetic and if the Gibbs measure mF for Γ is finite, then by Proposition 4.15, thediscrete time geodesic flow on ΓzGX1 is topologically mixing, hence by conjugation by Θ, thetopological Markov shift pΣ, σq is topologically mixing. This concludes the proof of Theorem5.1. l

When the length spectrum LΓ of Γ is different from Z, the topological Markov shift pΣ, σqconstructed above is not always topologically mixing. We now modify the above constructionin order to take care of this problem.

73 19/12/2016

Recall that X1 “ C ΛΓ and that GevenX1 is the space of geodesic lines ` P GX1 whose origin`p0q is at even distance from the basepoint x0 (we assume that x0 P X1), which is invariantunder the time-two discrete geodesic flow g2 and, when LΓ “ 2Z, under Γ.

Consider Aeven the alphabet consisting of the quintuples pf´, h´, f0, h`, f`q where thetriples pf´, h´, f0q and pf0, h`, f`q belong to A and opf0q is at even distance from theimage in Y “ ΓzX1 of the basepoint x0. Let Aeven “ pAeven, i,jqi,jPAeven be the transitionmatrix with line and column indices in Aeven such that for all i “ pf´, h´, f0, h`, f`q andj “ pf´˚ , h

´˚ , f

0˚ , h

`˚ , f

`˚ q, we have Aeven, i,j “ 1 if and only if f` “ f´˚ . We denote by

pΣeven, σevenq the associated topological Markov shift. We endow Σeven with the slightlymodified distance

devenpx, x1q “ e´

12

max

nPN : @ k P t´n, ..., nu, xk “ x1k

(

,

where x “ pxkqkPZ and x1 “ px1kqkPZ are in Σeven.We have a canonical injection inj : Σeven Ñ Σ sending the sequence pf´n , h´n , f0

n, h`n , f

`n qnPZ

to pe´n , hn, e`n qnPZ with, for every n P Z,

e´2n “ f´n , h2n “ h´n , e`2n “ f0

n, e´2n`1 “ f0

n, h2n`1 “ h`n , e`2n`1 “ f`n ,

By construction, inj is clearly a homeomorphism onto its image, and

ΘpΓzGevenX1q “ injpΣevenq .

If two sequences in Σeven coincide between ´n and n, then their images by inj coincidebetween ´2n and 2n. Conversely, if the images by inj of two sequences in Σeven coincidebetween ´2n´ 1 and 2n` 1, then these sequences coincide between ´n and n. Hence inj isbilipschitz, for the above distances.

Let us define Θeven “ inj´1 ˝Θ|ΓzGevenX1 : ΓzGevenX1 Ñ Σeven. The following diagram hencecommutes

ΓzGevenX1ΘevenÝÝÝÝÑ Σeven

§

§

đ

§

§

đ

inj

ΓzGX1 ΘÝÝÝÝÑ Σ ,

where the vertical map on the left hand side is the inclusion map.

Theorem 5.2. Assume that X1 “ C ΛΓ is a uniform tree without vertices of degree at most 2,that the Gibbs measure mF of Γ is finite, and that the length spectrum LΓ of Γ is equal to 2Z.Then the map Θeven : ΓzGevenX1 Ñ Σeven is a bilipschitz homeomorphism which conjugatesthe time-two discrete geodesic flow g2 and the shift σeven, and the topological Markov shiftpΣeven, σevenq is locally compact and topologically mixing.

Proof. The only claims that remains to be proven is the last one, which follows fromProposition 4.15, by conjugation. l

Let us now study the properties of the image by the coding map Θ of finite Gibbs measureson ΓzGX.

Let pµ˘x qxPV X be two Patterson densities on B8X for the pairs pΓ, F˘q, where as previouslyrF` “ rF , rF´ “ rF ˝ ι. Assume that the associated Gibbs measure mF on ΓzGX (using theconvention for discrete time of Section 4.3) is finite.

74 19/12/2016

Let us defineP “

1

mF Θ˚mF (5.4)

as the image of the Gibbs measure mF (whose support is ΓzGX1) by the homeomorphism Θ,normalised to be a probability measure. It is a probability measure on Σ, invariant under theshift σ.

Let pZnqnPZ be the random process classically associated with the full shift σ on Σ: it isthe random process on the Borel space Σ indexed by Z with values in the discrete alphabetA , where Zn : Σ Ñ A is the (continuous hence measurable) n-th projection pxkqkPN ÞÑ xnfor all n P Z.

The following result summarises the properties of the probability measure P. We start byrecalling and giving some notation used in this proposition.

For every admissible finite sequence w “ pap, . . . , aqq in A , where p ď q P Z, we denote‚ by rws “ rap, . . . , aqs “ tpxnqnPZ P Σ : @ n P tp, . . . , qu, xn “ an

(

the associatedcylinder in Σ,‚ by pw the associated geodesic edge path in X with length q ´ p ` 2 constructed in the

proof of Theorem 5.1 (see Equation (5.3)), with origin pw´ and endpoint pw`.

For every geodesic edge path α “ pfp´1, . . . , fqq in X1, we define (See Section 2.7 for thenotation, and the picture below)

B`αX1 “ BfqX1 and B´αX1 “ B fp´1X1 ,

andGαX “ t` P GX : `pp´ 1q “ opfp´1q and `pq ` 1q “ tpfqqu .

fqfp´1B`αX1B´αX1

α

We define a map Fsymb : Σ Ñ R by

Fsymbpxq “

ż tpe`0 q

ope`0 qF (5.5)

if x “ pxiqiPZ with x0 “ pe´0 , h0, e

`0 q. Note that Fsymb is locally constant (constant on each

cylinder of length 1 at time 0), hence continuous: for all pxnqnPZ, pynqnPZ P Σ, if x0 “ y0, thenFsymbpxq “ Fsymbpyq.

For instance, if F “ Fc is the potential associated with a system of conductances c :ΓzEX1 Ñ R (see Section 3.5), then

Fsymbpxq “ cpe`0 q .

Note that if c, c1 : ΓzEX1 Ñ R are cohomologous systems of conductances on ΓzEX1, then thecorresponding maps Fsymb, F

1symb : Σ Ñ R are cohomologous. Indeed if f : ΓzV X Ñ R is a

map such that c1peq ´ cpeq “ fptpeqq ´ fpopeqq for all e P ΓzEX, with G : Σ Ñ R the mapdefined by Gpxq “ fpope`0 qq if x “ pxiqiPZ with x0 “ pe

´0 , h0, e

`0 q, then G is locally constant,

hence continuous, and since tpe`0 q “ ope`1 q, we have, for every x P Σ,

F 1symbpxq ´ Fsymbpxq “ Gpσxq ´Gpxq .

75 19/12/2016

Definition 5.3. Let X be a locally finite simplicial tree. A nonelementary discrete subgroupΓ1 of AutpXq isMarkov-good if for every n P N´t0u and every geodesic edge path pe0, . . . , en`1q

in C ΛΓ1, we have

|Γ1e0 X ¨ ¨ ¨ X Γ1en | |Γ1en´1

X Γ1en X Γ1en`1| “ |Γ1e0 X ¨ ¨ ¨ X Γ1en`1

| |Γ1en´1X Γ1en | . (5.6)

Remark 5.4. (1) Note that Equation (5.6) is automatically satisfied if n “ 1 and that Γ1 isMarkov-good if Γ1 acts freely on X.

(2) A group action on a tree is 2-acylindrical6 if the stabiliser of any geodesic edge pathof length 2 is trivial. If Γ1 is 2-acylindrical on X, then Γ1 is Markov-good, since all groupsappearing in Equation 5.6 are trivial.

(3) If X has degrees at least 3 and if Γ1 is a noncocompact geometrically finite lattice of X,then Γ1 is not Markov-good.

Proof. (3) Since the quotient graph Γ1zX is infinite, the graph of groups Γ1zzX contains atleast one cuspidal ray. Consider a geodesic ray in X with consecutive edges pfnqnPN mappinginjectively onto this cuspiday ray, pointing towards its end. Their stabilisers in Γ1 are hencenondecreasing: we have Γ1fn Ă Γ1fn`1

for all n P N. By the finiteness of the volume, there existsn ě 3 such that Γ1fn´2

is strictly contained in Γ1fn´1. Since X has degrees at least 3, there

exists γ P Γ1 fixing tpfn´1q but not fixing fn´1. Let e0 “ f0, . . . , en´1 “ fn´1, en “ γ fn´1

and en`1 “ γ fn´2. Then pe0, . . . , en`1q is a geodesic edge path in X (equal to C ΛΓ1 since Γ1

is a lattice). Since Γ1e0X¨ ¨ ¨XΓ1en “ Γ1f0, Γ1en´1

XΓ1enXΓ1en`1“ Γ1fn´2

, Γ1e0X¨ ¨ ¨XΓ1en`1“ Γ1f0

,Γ1en´1

X Γ1en “ Γ1fn´1and |Γ1fn´2

| ‰ |Γ1fn´1|, the subgroup Γ1 is not Markov-good. l

Recall that a random process pZnqnPZ on pΣ,Pq is a Markov chain if and only if for allp ď q in Z and ap, . . . , aq, aq`1 in A , we have

PpZq`1 “ aq`1 |Zq “ aq, . . . , Zp “ apq “ PpZq`1 “ aq`1 |Zq “ aqq . (5.7)

Proposition 5.5. (1) For every admissible finite sequence w in A , we have

Pprwsq “µ´pw´pB´pwX

1q µ`pw`pB`pwX

1q eş

pw`pw´p rF´δq

|Γpw| mF

.

(2) The random process pZnqnPZ on pΣ,Pq is a Markov chain if and only if Γ is Markov-good.

(3) The measure P on the topological Markov shift Σ satisfies the Gibbs property with Gibbsconstant δ for the potential Fsymb.

It follows from the above Assertion (2) and from Remark 5.4 that when X has degreesat least 3 and Γ is a noncocompact geometrically finite lattice of X, then pZnqnPZ is nota Markov chain. The fact that codings of discrete time geodesic flows on trees might notsatisfy the Markov chain property had been noticed by Burger and Mozes around the timethe paper [BuM] was published.7 When proving the variational principle in Section 5.4 andthe exponential decay of correlations in Section 9.2, we will hence have to use tools that arenot using the Markov chain property.

6See for instance [Sel, GuL], which require other minor hypothesis that are not relevant here.7Personal communication.

76 19/12/2016

Proof. (1) Let w “ pap, . . . , aqq where p ď q P Z, be an admissible finite sequence in A . Bythe construction of Θ, the preimage Θ´1prwsq is equal to the image ΓG

pwX1 of GpwX1 in ΓzGX1.

Hence, since Γpw is the stabiliser of G

pwX1 in Γ,

Pprwsq “1

mF mF pΓG

pwX1q “1

|Γpw| mF

rmF pGpwX1q .

In the expression of rmF given by Equation (4.10), let us use as basepoint x0 the origin pw´ ofthe edge path pw, and note that all elements of G

pwX1 pass through pw´ at time t “ p ´ 1, sothat by invariance of rmF under the geodesic flow , we have

rmF pGpwX1q “

ż

`PGpwX1

drmF pg1´p`q “

ż

`´PB´pw`

X1

ż

``PB`pw`

X1dµ´

pw´p`´qdµ

`pw´p``q

“ µ´pw´pB´pw`

X1q µ`pw´pB`pwX

1q “ µ´pw´pB´pwX

1q µ`pw`pB`pwX

1q eş

pw`pw´pF´δq

,

where this last equality follows by Equations (4.2) and (3.8) with x “ pw´ and y “ pw`, sincefor every `` P B`

pw`X1, we have pw` P r pw`, ``r .

(2) Let us fix p ď q in Z and ap, . . . , aq, aq`1 in A , and let us verify Equation (5.7). We mayassume that α “ pap, . . . , aq, aq`1q is an admissible sequence. Let α˚ “ pap, . . . , aqq, which isalso an admissible sequence. Let us consider

Qα “PpZq`1 “ aq`1 |Zq “ aq, . . . , Zp “ apq

PpZq`1 “ aq`1 |Zq “ aqq“

Pprap, . . . , aq`1sq PpraqsqPprap, . . . , aqsq Ppraq, aq`1sq

.

Let us replace each one of the four terms in this ratio by its value given by Assertion (1).Since B´

pαX1 “ B

´

xα˚X1, B`

pαX1 “ B

`

aq ,aq`1X1, B`

xα˚X1 “ B`

xaqX1 and B´

paqX1 “ B´

aq ,aq`1X1, all Patterson

measure terms cancel. Denoting by y1 the common origin of pα and xα˚, by y2 the commonorigin of paq and aq, aq`1, by y3 the common endpoint of paq and xα˚, and by y4 the commonendpoint of aq, aq`1 and pα, we thus have by Assertion (1)

Qα “|Γ

xα˚ | |Γ aq ,aq`1|

|Γpα| |Γxaq |

eşy4y1p rF´δq

eşy3y2p rF´δq

eşy3y1p rF´δq

eşy4y2p rF´δq

.

Since y1, y2, y3, y4 are in this order on ry1, y4s, we have

Qα “|Γ

xα˚ | |Γ aq ,aq`1|

|Γpα| |Γxaq |

.

Since every geodesic edge path of length n ` 1 at least 3 in X1 defines an admissiblesequence of length n at least 2 in A , by Equation (5.6), we have Qα “ 1 for every admissiblesequence α in A if and only if Γ is Markov-good.

(3) Let E be a finite subset of the alphabet A , and let w “ pap, . . . , aqq with p ď q in Z andadmissible sequence in A such that ap, aq P E. By Assertion (1), we have

Pprwsq “µ´pw´pB´pwX

1q µ`pw`pB`pwX

1q eş

pw`pw´p rF´δq

|Γpw| mF

.

77 19/12/2016

Since ap, aq are varying in the finite subset E of A , the first and last edges of pw vary amongstthe images under elements of Γ of finitely many edges of X. Since w is admissible, theopen sets B˘

pwX1 are nonempty subsets of ΛΓ, hence they have positive Patterson measures.

Furthermore, the quantities µ˘pw˘pB˘pwX

1q are invariant under the action of Γ on the first/lastedge of pw. Hence there exists c1 ě 1 depending only on E such that 1 ď |Γ

pw| ď |Γ pw´ | ď c1

and 1c1ď µ˘

pw˘pB˘pwX

1q ď c1.Note that the length of pw is equal to q ´ p` 2. Therefore

e´δ

c31 mF

e´δpq´p`1q`

ş

pw`pw´

rFď Pprwsq ď

e´δ c21

mF e´δpq´p`1q`

ş

pw`pw´

rF.

If pw “ pfp´1, fp, . . . , fqq and x P rws, we have by the definition of Fsymb

ż

pw`

pw´

rF “

qÿ

i“p´1

ż tpfiq

opfiq

rF “

ż tpfp´1q

opfp´1q

rF `

qÿ

i“p

Fsymbpσipxqq .

Since F is continuous and opfp´1q remains in a finite subset of V X1, there exists c2 ą 0

depending only on E such that | rF pvq| ď c2 for every v P T 1X with πpvq P ropfp´1q, tpfp´1qs.Hence |

ştpfp´1q

opfp´1qrF | ď c2, and Assertion (3) of Proposition 5.5 follows (see Equation (5.1) for

the definition of the Gibbs property). l

Again in order to consider the case when the length spectrum LΓ of Γ is 2Z, we define

Peven “1

pmF q| ΓzGevenX1pΘevenq˚pmF q| ΓzGevenX1 ,

and pZeven, nqnPZ the random process associated with the full shift σeven on Σeven, with Zeven, n :Σeven Ñ Aeven the n-th projection for every n P Z.

By a proof similar to the one of Proposition 5.5, we have the following result. We definea map Fsymb, even : Σeven Ñ R by

Fsymb, evenpxq “

ż tpf`0 q

opf00 q

F (5.8)

if x “ pxiqiPZ with x0 “ pf´0 , h

´0 , f

00 , h

`0 , f

`0 q. As previously, Fsymb, even is locally constant,

hence continuous.

Proposition 5.6. The measure Peven on the topological Markov shift Σeven satisfies the Gibbsproperty with Gibbs constant δ for the potential Fsymb, even. l

Again, if Γ is a noncocompact geometrically finite lattice of X and X1 has degrees at least3, then pZeven, nqnPZ is not a Markov chain.

5.3 Coding continuous time geodesic flows on metric trees

Let pX, λq be a locally finite metric tree without terminal vertices, with X “ |X|λ its geometricrealisation. Let Γ be a nonelementary discrete subgroup of AutpX, λq, and let rF : T 1X Ñ Rbe a potential for Γ. Let X 1 “ C ΛΓ, which is the geometric realisation |X1|λ of a metric

78 19/12/2016

subtree pX1, λq. Let pµ˘x qxPV X be two Patterson densities on B8X for the pairs pΓ, F˘q, andassume that the associated Gibbs measure mF is finite. We also assume in this Section thatthe lengths of the edges of pX1, λq have a finite upper bound (which is in particular the caseif pX1, λq is uniform). They have a positive lower bound by definition (see Section 2.7).

In this Section, we prove that the continuous time geodesic flow on ΓzGX 1 is isomorphicto a suspension of a transitive (two-sided) topological Markov shift on a countable alphabet,by an explicit construction that will be useful later on to study the variational principle (seeSection 5.4) and rates of mixing (see Section 9.3). Since we are only interested in the supportof the Gibbs measures, we will only give such a description for the geodesic flow on the non-wandering subset ΓzGX 1 of ΓzGX. The same construction works with the full space ΓzGX,but the resulting Markov shift is then not necessarily transitive.

We start by recalling (see for instance [BrinS, §1.11]) the definitions of the suspension ofa (invertible) discrete time dynamical system and of the first return map on a cross-section ofa continuous time dynamical system, which allow to pass from transformations to flows andback, respectively.

Let pZ, µ, T q be a metric space Z endowed with an homeomorphism T and a T -invariant(Borel, positive) measure µ. Let r : Z Ñ s0,`8r be a continuous map, such that for allz P Z, the subset trpTnzq : n P Nu Y t´ rpT´pn`1qzq : n P Nu is discrete in R. Then thesuspension (or also special flow) over pZ, µ, T q with roof function r is the following continuoustime dynamical system pZr, µr, pT

trqtPRq :

‚ The space Zr is the quotient topological space pZ ˆ Rq „ where „ is the equivalencerelation on Z ˆ R generated by pz, s ` rpzqq „ pTz, sq for all pz, sq P Z ˆ R. We denote byrz, ss the equivalence class of pz, sq. Note that

F “ tpz, sq : z P Z, 0 ď s ă rpzqu

is a measurable strict fundamental domain for this equivalence relation. We endow Zr withthe Bowen-Walters distance, see [BowW] and particularly the appendix in [BarS].‚ For every t P R, the map T tr : Zr Ñ Zr is the map rz, ss ÞÑ rz, s ` ts. Equivalently,

when pz, sq P F and t ě 0, then T trprz, ssq “ rTnz, s1s where n P N and s1 P R are such that

t` s “n´1ÿ

i“0

rpT izq ` s1 and 0 ď s1 ă rpTnzq .

‚ With ds the Lebesgue measure on R, the measure µr is the pushforward of the restrictionto F of the product measure dµ ds by the restriction to F of the canonical projection pZ ˆRq Ñ Zr.

Note that pT trqtPR is indeed a continuous 1-parameter group of homeomorphisms of Zr,preserving the measure µr. The measure µr is finite if and only if

ş

Z r dµ is finite, since

µr “

ż

Zr dµ .

We will denote by pZ, µ, T qr the continuous time dynamical system pZr, µr, pTtrqtPRq thus

constructed.

Conversely, let pZ, µ, pφtqtPRq be a metric space Z endowed with a continuous 1-parametergroup of homeomorphisms pφtqtPR, preserving a (Borel, positive) measure µ. Let Y be a

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cross-section of pφtqtPR, that is a closed subspace of Z such that for every z P Z, the settt P R : φtpzq P Y u is nonempty and discrete. Let τ : Y Ñ s0,`8r be the (continuous) firstreturn time on the cross-section Y : for every y P Y ,

τpyq “ mintt ą 0 : φtpyq P Y u .

Let φY : Y Ñ Y be the (homeomorphic) first return map to (or Poincaré map of) the cross-section Y , defined by

φY : y ÞÑ φτpyqpyq .

By the invariance of µ under the flow pφtqtPR, the restriction of µ to

tφtpyq : y P Y, 0 ď t ă τpyqu

disintegrates8 by the (well-defined) map φtpyq ÞÑ y over a measure µY on Y , which is invariantunder the first return map φY :

dµpφtpyqq “ dt dµY pyq .

Note that if r has a positive lower bound and if µ is finite, then µY is finite, since

µ ě µY inf r .

Hence pY, µY , φY q is a discrete time dynamical system.

Recall that an isomorphism from a continuous time dynamical system pZ, µ, pφtqtPRq toanother one pZ 1, µ1, pφ1tqtPRq is an homeomorphism between the underlying spaces preservingthe underlying measures and commuting with the underlying flows.

Example 5.7. If pZ, µ, T q and pZ 1, µ1, T 1q are (invertible) discrete time dynamical systems,endowed with roof functions r : Z Ñ s0,`8r and r1 : Z 1 Ñ s0,`8r respectively, if θ : Z Ñ Z 1

is a measure preserving homeomorphism commuting with the transformations T and T 1 (thatis, θ˚µ “ µ1, θ ˝ T “ T 1 ˝ θ) and such that

r1 ˝ θ “ r ,

then the mapp

θ : Zr Ñ Z 1r1 defined by rz, ss ÞÑ rθpzq, ss is an isomorphim between thesuspensions pZ, µ, T qr and pZ 1, µ1, T 1qr1 .

It is well known (see for instance [BrinS, §1.11]) that the above two constructions areinverses one to another, up to isomorphism. In particular, we have the following result.

Proposition 5.8. The suspension pY, µY , φY qτ over pY, µY , φY q with roof function τ is iso-morphic to pZ, µ, pφtqtPRq by the map fY : ry, ss ÞÑ φsy. l

In order to describe the continuous time dynamical system`

ΓzGX 1, mFmF

, pgtqtPR˘

as asuspension over a topological Markov shift, we will start by describing it as a suspensionof the discrete time geodesic flow on ΓzGX1. Note that the Patterson densities and Gibbs

8with conditional measure on the fiber tφtpyq : 0 ď t ă τpyqu over y P Y the image of the Lebesguemeasure on r0, τpyqr by t ÞÑ φtpyq

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measures depend not only on the potential, but also on the lengths of the edges.9 We henceneed to relate precisely the continuous time and discrete time situations, and we will use inthis Section the left exponent 7 to indicate a discrete time object whenever needed.

For instance, we set 7X 1 “ |X1|1 and we denote by p7gtqtPZ the discrete time geodesicflow on ΓzGX1. Note that X 1 and 7X 1 are equal as topological spaces (but not as metricspaces). The boundaries at infinity of X 1 and 7X 1, which coincide with their spaces of endsas topological spaces (by the assumption on the lengths of the edges), are hence equal anddenoted by B8X.

We may assume by Section 3.5 that the potential rF : T 1X Ñ R is the potential rFcassociated with a system of conductances rc on the metric tree pX, λq for Γ. Let δc “ δFc . Wedenote by 7rc : EXÑ R the Γ-invariant system of conductances

7rc : e ÞÑ pcpeq ´ δcqλpeq (5.9)

on the simplicial tree X for Γ, by rF7c : T 1p 7Xq Ñ R its associated potential, and by 7c :ΓzEXÑ R and F7c : ΓzT 1p 7Xq Ñ R their quotient maps.

Note that the inclusion morphism AutpX, λq Ñ AutpXq is a homeomorphism onto its image(for the compact-open topologies), by the assumption on the lengths of the edges, hence thatΓ is also a nonelementary discrete subgroup of AutpXq.

Now, let pΣ, σ,Pq be the (two-sided) topological Markov shift conjugated to the discretetime geodesic flow

`

ΓzGX1, 7g1,mF7cmF7c

˘

by the bilipschitz homeomorphism Θ : ΓzGX1 Ñ Σ of

Theorem 5.1 (where the potential F is replaced by F7c). Let r : Σ Ñ s0,`8r be the map

r : x ÞÑ λpe`0 q (5.10)

if x “ pxnqnPZ P Σ and x0 “ pe´0 , h0, e

`0 q P A . This map is locally constant, hence continuous

on Σ, and has a positive lower bound, since the lengths of the edges of pX1, λq have a positivelower bound.

Theorem 5.9. Assume that the lengths of the edges of pX1, λq have a finite upper bound,and that the Gibbs measure mF is finite. Then there exists a ą 0 such that the continuoustime dynamical system

`

ΓzGX 1, mFmF

, pgtqtPR˘

is isomorphic to the suspension pΣ, σ, aPqr overpΣ, σ, aPq with roof function r, by a bilipschitz homeomorphism Θr : ΓzGX 1 Ñ Σr.

Proof. LetY “ t` P ΓzGX 1 : `p0q P ΓzV Xu .

Then the (closed) subset Y of ΓzGX 1 is a cross-section of the continuous time geodesic flowpgtqtPR, since every orbit meets Y and since the lengths of the edges of pX1, λq have a positivelower bound. Let τ : Y Ñ s0,`8r be the first return time, let µY be the measure on Y(obtained by disintegrating mF

mF ), and let gY : Y Ñ Y be the first return map associated

with this cross-section Y .9The fact that the Patterson densities could be singular one with respect to another when the metric varies

is a well known phenomenon, even when the potential vanishes. See for instance Kuusalo’s theorem [Kuu]saying that the Patterson densities on the boundary at infinity of the real hyperbolic plane of two cocompactmarked Fuchsian groups are absolutely continuous one with respect to the other if and only if the markedFuchsian groups are conjugated, and the extension of this result in [HeP1]. See also the result of [KaN]which parametrises the Culler-Vogtmann space using Patterson densities for cocompact and free actions offree groups on metric trees.

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We have a natural reparametrisation map R : Y Ñ ΓzGX1, defined by ` ÞÑ 7`, where7`pnq “ pgnY `qp0q is the n-th passage of ` in V X, for every n P Z. Since there exists m,M ą 0such that λpEXq Ă rm,M s, the map R is a bilipschitz homeomorphism. It commutes withthe first return map gY and the discrete time geodesic flow on ΓzGX1:

R ˝ gY “7g1 ˝R .

The main point of this proof is the following result relating the measures µY and mF7c.

Lemma 5.10. (1) The family pµ˘x qxPV X is a Patterson density for pΓ, F7cq on the boundaryat infinity of the simplicial tree X1, and the critical exponent δ7c of 7c is equal to 0.

(2) We have R˚ µYµY

“mF7cmF7c

.

Proof. (1) By the definition of the potential associated with a system of conductances (seeSection 3.5), for every x, y P V X1, if pe1, . . . , enq is the edge path in X with ope1q “ x andtpenq “ y, then (noting that the integrals along paths depend on the lengths of the edges, thefirst one below being in X 1, the second one in 7X 1)

ż y

xpĂFc ´ δcq “

nÿ

i“1

prcpeiqλpeiq ´ δcλpeiqq “

ż y

x

ĂF7c . (5.11)

Let us denote (see Section 3.3) by

7Qpsq “ QΓ, F7c, x, y“

ÿ

γPΓ

eşγyx p

rF7c´sq

andQpsq “ QΓ, Fc´δc, x, ypsq “

ÿ

γPΓ

eşγyx p

rFc´δc´sq

the Poincaré series for the simplicial tree with potential rF7c and for the metric tree withnormalised potential rFc ´ δc, respectively. We hence have 7Qpsq ď Qp sM q ă `8 if s ą 0 and7Qpsq ě Qp sM q “ `8 if s ă 0. Thus the critical exponent δ7c of pΓ, F7cq for the simplicialtree X1 is equal to 0, hence F7c is a normalised potential.

By the definition (see Section 3.4) of the Gibbs cocycles (which uses the normalised po-tential), Equation (5.11) also implies that the Gibbs cocycles C˘ and 7C˘ for pΓ, Fcq andpΓ, F7cq respectively coincide on B8X ˆ V X ˆ V X. Thus by Equations (4.1) and (4.2), thefamily pµ˘x qxPV X is indeed a Patterson density for pΓ, F7cq: for all γ P Γ and x, y P V X, andfor (almost) all ξ P B8X,

γ˚µ˘x “ µ˘γx and

dµ˘xdµ˘y

pξq “ e´7C˘ξ px, yq .

(2) We may hence choose these families pµ˘x qxPV X in order to define the Gibbs measure m7c

associated with the potential F7c on ΓzGX. Note that since we will prove that m7c is finite,the normalised measure m7c

m7cis independent of this choice (see Corollary 4.6).

Let rY “ t` P GX 1 : `p0q P V X1u be the (Γ-invariant) lift of the cross-section Y to GX 1,let rR : rY Ñ GX1 be the lift of R, mapping a geodesic line ` P rY to a discrete geodesic line

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7` obtained by reparametrisation, and let õY be the measure on rY whose induced measureon Y “ ΓzrY is µY . We have a partition of rY into the closed-open subsets rYx “ t` P GX 1 :`p0q “ xu as x varies in V X1.

Let us fix x P V X. By the definition of µY as a disintegration of mFcmFc

with respect to thecontinuous time, by lifting to GX 1, by using Hopf’s parametrisation with respect to x andEquation (4.3) with x0 “ x , we have for every ` P rYx,

dõY p`q “1

mFcdµ´x p`´q dµ

`x p``q .

Note that `p0q “ 7`p0q, `´ “ 7`´, `` “ 7`` since the reparametrisation does not change theorigin nor the two points at infinity. Hence by Assertion (1), we have

rR˚põY q “1

mFcrmF7c

.

As µY is a finite measure since τ has a positive lower bound, this implies that mF7cis finite.

By renormalizing as probability measures, this proves Assertion (2). l

Let a “ µY ą 0, so that by Lemma 5.10 (2) we have R˚µY “a m7cm7c

. Letp

r : ΓzGX1 Ñs0,`8s be the map p

r : Γ` ÞÑ λ`

`pr0, 1sq˘

given by the length for λ of the first edge followed by a discrete geodesic line ` P GX1. Notethat

p

r is locally constant, hence continuous, and thatp

r is a roof function for the discrete timedynamical system pΓzGX1, 7g1q. Also note that

p

r ˝R “ τ and r ˝Θ “

p

r

by the definitions of τ and r.Let us finally define Θr : ΓzGX 1 Ñ Σr as the compositions of the following three maps

pΓzGX 1,mF

mF , pgtqtPRq

f ´1YÝÑ pY, µY , gY qτ

p

RÝÑ pΓzGX1,

a m7c

m7c, 7g1q

p

r

p

ΘÝÑ pΣ, aP, σqr , (5.12)

where the first one is the inverse of the tautological isomorphism given by Proposition 5.8and the last two ones, given by Example 5.7, are the isomorphisms

p

R andp

Θ of continuoustime dynamical systems obtained by suspensions of the isomorphisms R and Θ of discretetime dynamical systems. It is easy to check that Θr is a bilipschitz homeomorphism, usingthe following description of the Bowen-Walters distance, see for instance [BarS, Appendix].

Proposition 5.11. Let pZ, µ, T qr be the suspension over an invertible dynamical system suchthat T is a bilipschitz homeomorphism, with roof function r having a positive lower boundand a finite upper bound. Let dBW : Zr ˆ Zr Ñ R be the map10 defined (using the canonicalrepresentatives) by

dBWprx, ss, rx1, s1sq “

mintdpx, x1q ` |s´ s1|, dpTx, x1q ` rpxq ´ s` s1, dpx, Tx1q ` rpyq ` s´ s1u

10The map dBW is actually not a distance, but may replace the Bowen-Walters true distance when workingup to multiplicative constants or bilipschitz homeomorphisms.

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Then there exists a constant CBW ą 0 such that the Bowen-Walters distance d on Σr satisfies

1

CBWdBW ď d ď CBW dBW . l

This concludes the proof of Theorem 5.9. l

5.4 The variational principle for metric and simplicial trees

In this Section, we assume that X is the geometric realisation of a locally finite metric treewithout terminal vertices pX, λq (respectively of a locally finite simplicial tree X withoutterminal vertices). Let Γ be a nonelementary discrete subgroup of AutpX, λq (respectivelyAutpXq).

We relate in this Section the Gibbs measures11 to the equilibrium states (see the definitionsbelow) for the continuous time geodesic flow on ΓzGX (respectively for the discrete timegeodesic flow on ΓzGX).

When X is a Riemannian manifold with pinched negative curvature such that the deriva-tives of the sectional curvature are uniformly bounded, and when the potential is Hölder-continuous, the analogs of the results of this Section are due to [PauPS, Thm. 6.1]. Theirproofs generalise the proofs of Theorems 1 and 2 of [OP], with ideas and techniques goingback to [LedS]. When Y is a compact locally CATp´1q-space, a complete statement aboutexistence, uniqueness and Gibbs property of equilibrium states for any Hölder continuouspotential is given in [ConLT].

The proof of the metric tree case will rely strongly (via the suspension process describedin Section 5.3) on the proof of the simplicial tree case, hence we start by the latter.

The simplicial tree case.

Let X be a locally finite simplicial tree without terminal vertices, with geometric realisationX “ |X|1. Let Γ be a nonelementary discrete subgroup of AutpXq and let rc : EX Ñ R be asystem of conductances for Γ on X. Let rFc : T 1X Ñ R be its associated potential, and let δcbe the critical exponent of c.

We define a map rFc : GXÑ R by

rFcp`q “ rc pe`0 p`qq “

ż tpe`0 p`qq

ope`0 p`qq

rFc

for all ` P GX, where e`0 p`q is the edge of X in which ` enters at time t “ 0. This map islocally constant, hence continuous, and it is Γ-invariant, hence it induces a continuous mapFc : ΓzGXÑ R.12

The following result proves that the Gibbs measure of pΓ, Fcq for the discrete time geodesicflow on ΓzGX is an equilibrium state for the potential Fc. We start by recalling the definitionof an equilibrium state,13 see also [Bowe2, Rue3].

11See the definition in Sections 4.2 and 4.3.12See after the proof of Theorem 5.12 for a comment on cohomology classes.13This definition is given for transformations and not flows, and for possibly unbounded potentials, contrarily

to the one the Introduction.

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Let Z be a locally compact topological space, let T : Z Ñ Z be a homeomorphism, andlet φ : Z Ñ R be a continuous map. Let Mφ be the set of Borel probability measures mon Z, invariant under the transformation T , such that the negative part maxt0,´φu of φ ism-integrable. Let hmpT q be the (metric) entropy of the transformation T with respect tom P Mφ. The metric pressure for the potential φ of a measure m P Mφ is

Pφpmq “ hmpT q `

ż

Zφdm .

The pressure of the potential φ is

Pφ “ supmPMφ

Pφpmq .

A measure m0 P Mφ is an equilibrium state for the potential φ if Pφpm0q “ Pφ.

Theorem 5.12 (The variational principle for simplicial trees). Let X,Γ,rc be as above. Assumethat δc ă `8 and that there exists a finite Gibbs measure mc for Fc such that the negativepart of the potential Fc is mc-integrable. Then mc

mcis the unique equilibrium state for the

potential Fc under the discrete time geodesic flow on ΓzGX, and the pressure of Fc coincideswith the critical exponent δc of c.

In order to prove this result, using the coding of the discrete time geodesic flow given inSection 5.2, the main tool is the following result of J. Buzzi in symbolic dynamics, buildingon works of Sarig and Buzzi-Sarig, whose proof is given in the Appendix.

Let σ : Σ Ñ Σ be a two-sided topological Markov shift14 with (countable) alphabet Aand transition matrix A, and let φ : Σ Ñ R be a continuous map.

For every n P N, we denote by

varn φ “ supx, y PΣ

@ i P t´n, ..., nu, xi“yi

|φpxq ´ φpyq|

the n-variation of φ. For instance, if φpxq depends only on x0, then varn φ “ 0 (and henceř

nPN n varn φ “ 0 converges).A weak Gibbs measure for φ with Gibbs constant C “ Cpmq P R is a σ-invariant Borel

measure m on Σ such that for every a P A , there exists ca ě 1 such that for all n P N´ t0uand x P ras such that σnpxq “ x, we have

1

caď

mprx0, . . . , xn´1sq

e´Cpmqn`řn´1i“0 φpσixq

ď ca . (5.13)

Theorem 5.13 (J. Buzzi, see Corollary A.5). Let pΣ, σq be a two-sided transitive topologicalMarkov shift on a countable alphabet A and let φ : Σ Ñ R be a continuous map such thatř

nPN n varn φ converges. Let m be a weak Gibbs measure for φ on Σ with Gibbs constantCpmq, such that

ş

φ´ dm ă `8. Then the pressure of φ is finite, equal to Cpmq, and m isthe unique equilibrium state. l

14See Section 5.1 for definitions.

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Proof of Theorem 5.12. In Section 5.2, we constructed a transitive topological Markovshift pΣ, σq on a countable alphabet A and a homeomorphism Θ : ΓzGX1 Ñ Σ which conju-gates the time-one discrete geodesic flow g1 on the nonwandering subset ΓzGX1 of ΓzGX andthe shift σ on Σ (see Theorem 5.1). Let us define a potential Fc, symb : Σ Ñ R by

Fc, symbpxq “ cpe`0 q (5.14)

if x “ pxiqiPZ with x0 “ pe´0 , h0, e

`0 q. Note that this potential is the one denoted by Fsymb in

Equation (5.5), when the potential F on T 1X is replaced by Fc. By the construction of Θ,we have

Fc, symb ˝Θ “ Fc . (5.15)

Note that all probability measures on ΓzGX invariant under the discrete time geodesic floware supported on the nonwandering set ΓzGX1. The pushforward of measures Θ˚ hence givesa bijection from the space MFc of g1-invariant probability measures on ΓzGX for which thenegative part of Fc is integrable to the space MFc, symb

of σ-invariant probability measures onΓzGX for which the negative part of Fc, symb is integrable. This bijection induces a bijectionbetween the subsets of equilibrium states.

Since Fc, symbpxq depends only on x0 for every x P Σ, the seriesř

nPN n varn Fc, symb

converges.By definition (see Equation (5.4)), the measure P is the pushforward of mFc

mFcby Θ,

hence is a σ-invariant probability measure on ΓzGX for which the negative part of Fc, symb isintegrable, by the assumption of Theorem 5.12. By Proposition 5.5 (3), the measure P on Σsatisfies the Gibbs property with Gibbs constant δc for the potential Fc, symb, hence15 satisfiesthe weak Gibbs property with Gibbs constant δc. Theorem 5.12 then follows from Theorem5.13. l

Remark. It follows from Equation (5.15), from the remark above Proposition 5.5 and fromthe fact that Θ˝g1 “ σ˝Θ, that if c, c1 : ΓzEX1 Ñ R are cohomologous systems of conductanceson ΓzEX1, then the corresponding maps Fc,Fc1 : ΓzGX1 Ñ R are cohomologous: there existsa continuous map G : ΓzGX1 Ñ R such that for every ` P ΓzGX1,

Fc1p`q ´ Fcp`q “ Gpg1`q ´Gp`q .

The metric tree case.

Let pX, λq be a locally finite metric tree without terminal vertices with geometric realisationX “ |X|λ, let Γ be a nonelementary discrete subgroup of AutpX, λq and let rc : EXÑ R be asystem of conductances for Γ on X. Let rFc : T 1X Ñ R be its associated potential (see Section3.5), and let δc “ δFc be the critical exponent of c.

Recall16 that we have a canonical projection GX Ñ T 1X which associates to a geodesicline ` its germ v` at its footpoint `p0q. Let rF6c : GX Ñ R be the Γ-invariant map obtained byprecomposing the potential rFc : T 1X Ñ R with this canonical projection:

rF6c : ` ÞÑ rFcpv`q .

15For every a P A , for the constant ca required by the definition of the weak Gibbs property in Equation(5.13), take the constant CE given by the definition (see Equation (5.1)) of the Gibbs property with E “ tau.

16See Section 2.4.

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Let F6c : ΓzGX Ñ R be its quotient map, which is continuous, as a composition of continuousmaps.

The following result proves that the Gibbs measure of pΓ, Fcq for the continuous timegeodesic flow on ΓzGX is an equilibrium state for the potential F6c. We start by recalling thedefinition of an equilibrium state for a possibly unbounded potential under a flow.17

Given pZ, pφtqtPRq a topological space endowed with a continuous one-parameter group ofhomeomorphisms and ψ : Z Ñ R a continuous map (called a potential), let Mψ be the set ofBorel probability measures m on Z invariant under the flow pφtqtPR, such that the negativepart of ψ ism-integrable. Let hmpφ1q be the (metric) entropy of the geodesic flow with respectto m P Mψ. The metric pressure for ψ of a measure m P Mψ is

Pψpmq “ hmpφ1q `

ż

Zψ dm .

The pressure of the potential ψ is

Pψ “ supmPMψ

Pψpmq .

An element m P Mψ is an equilibrium state for ψ if the least upper bound defining Pψ isattained on m.

Note that if ψ1 is another potential cohomologous to ψ, that is, if there exists a continuousmap G : Z Ñ R, differentiable along every orbit of the flow, such that ψ1pxq ´ ψpxq “ddt |t“0

Gpgtxq, then Mψ1 “ Mψ, for every m P Mψ, we have Pψ1pmq “ Pψpmq, Pψ1 “ Pψ andthe equilibrium states for ψ1 are exactly the equilibrium states for ψ.

Theorem 5.14 (The variational principle for metric trees). Let pX, λq,Γ,rc be as above. As-sume that the lengths of the edges of pX, λq have a finite upper bound.18 Assume that δc ă `8and that there exists a finite Gibbs measure mc for Fc such that the negative part of the poten-tial F6c is mc-integrable. Then mc

mcis the unique equilibrium state for the potential F6c under

the continuous time geodesic flow on ΓzGX, and the pressure of F6c coincides with the criticalexponent δc of c.

Using the description of the continuous time dynamical system`

ΓzGX 1, mcmc

, pgtqtPR˘

asa suspension over a topological Markov shift (see Theorem 5.9), this statement reduces towell-known techniques in the thermodynamic formalism of suspension flows, see for instance[IJT], as well as [BarI, Kemp, IJ, JKL]. Our situation is greatly simplified by the fact thatour roof function has a positive lower bound and a finite upper bound, and that our symbolicpotential is constant on the 1-cylinders tx P Σ : x0 “ au for a in the alphabet.

Proof. Since finite measures invariant under the geodesic flow on ΓzGX are supported on itsnonwandering set, up to replacing X by X 1 “ C ΛΓ, we assume that X “ X 1.

Since equilibrium states are unchanged up to adding a constant to the potential, underthe assumptions of Theorem 5.14, let us prove that mc

mcis the unique equilibrium state for

the potential F6c´δc under the continuous time geodesic flow on ΓzGX, and that the pressureof F6c ´ δc vanishes. The last claim of Theorem 5.14 follows, since

PF6c´ δc “ PF6c´δc

.

17This requires only minor modifications to the definition given in the Introduction for bounded potentials.18They have a positive lower bound by definition, see Section 2.7.

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We refer to the paragraphs before the statement of Theorem 5.9 for the definitions of‚ the system of conductances 7rc for Γ on the simplicial tree X,‚ the (two-sided) topological Markov shift pΣ, σ,Pq on the alphabet A , conjugated to the

discrete time geodesic flow`

ΓzGX, 7g1,mF7cmF7c

˘

by the homeomorphism Θ : ΓzGXÑ Σ,

‚ the roof function r : Σ Ñ s0,`8r

‚ and the suspension pΣ, σ, aPqr “ pΣr, pσtrqtPR, aPrq over pΣ, σ, aPq with roof function r,

conjugated to the continuous time geodesic flow`

ΓzGX, mcmc

, pgtqtPR˘

by the homeomorphismΘr : ΓzGX Ñ Σr defined at the end of the proof of Theorem 5.9. We will always (uniquely)represent the elements of Σr as rx, ss with x P Σ and 0 ď s ă rpxq.

We denote by F6c, symb : Σr Ñ R the potential defined by

F6c, symb “ F6c ˝Θ´1r , (5.16)

which is continuous, as a composition of continuous maps. The key technical observation inthis proof is the following one.

Lemma 5.15. For every x P Σ, we have F7c, symbpxq “şrpxq0 pF6c, symb ´ δcqprx, ssq ds. For

every x P Σ, the sign of F6c, symbprx, ssq is constant on s P r0, rpxqs.

Proof. Let x “ pxnqnPZ P Σ and x0 “ pe´0 , h0, e

`0 q P A . By the definition of the first return

time r in Equation (5.10), we have in particular

rpxq “ λpe`0 q .

By Equation (5.14) and by the definition of 7rc in Equation (5.9), we have

F7c, symbpxq “7cpe`0 q “ pcpe

`0 q ´ δcqλpe

`0 q .

Using in the following sequence of equalities respectively‚ the definitions of the potential F6c, symb in Equation (5.16) and of the suspension flow

pσtrqtPR,‚ the fact that this suspension flow is conjugated to the continuous time geodesic flow by

Θr,‚ the definition of Θr using the reparametrisation map R of continuous time geodesic

lines with origin on vertices to discrete geodesic lines,‚ the definition of the potential F6c,‚ the fact that the first edge followed by the discrete geodesic line Θ´1x, hence by the

geodesic line R´1Θ´1x, is e`0 and the relation between the potential Fc associated with c andc (see Proposition 3.11),we have

ż rpxq

0F6c, symbprx, ssq ds “

ż rpxq

0F6c`

Θ´1r σsrrx, 0s

˘

ds “

ż rpxq

0F6c`

gsΘ´1r rx, 0s

˘

ds

“

ż rpxq

0F6c`

gsR´1Θ´1x˘

ds “

ż λpe`0 q

0Fc`

vgsR´1Θ´1x

˘

ds

“ cpe`0 qλpe`0 q .

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Sinceşrpxq0 δc ds “ δc λpe

`0 q, the first claim of Lemma 5.15 follows. The second claim

follows by the definition of the potential Fc associated with c, see Equation (3.11). l

By Equation (5.16), the pushforwards of measures by the homeomorphism Θr, whichconjugates the flows pgtqtPR and pσtrqtPR, is a bijection from MF6c

to MF6c, symb, such that

PF6c, symbppΘrq˚mq “ PF6c

pmq

for every m P MF6c. In particular, we only have to prove that pΘrq˚

mcmc

“ aPr is the unique

equilibrium state for the potential F6c, symb ´ δc under the suspension flow pσtrqtPR, and thatthe pressure of F6c, symb ´ δc vanishes.

The uniqueness follows for instance from [IJT, Theo. 3.5], since the roof function r islocally constant and the potential g “ F6c, symb is such that the map19 from Σ to R defined by

x ÞÑşrptq0 gprx, ssq ds is locally Hölder-continuous by Lemma 5.15 and since F7c, symb is locally

constant.

Let us now relate the σ-invariant measures on Σ with the pσtrqtPR-invariant measures onΣr. Recall that we denote the Lebesgue measure on R by ds and the points in Σr by rx, sswith x P Σ and 0 ď s ď rpxq.

Lemma 5.16. The map S : MF7c, symbÑ MF6c, symb

defined by

dSpmqprx, ssq “1

ş

Σ r dmdµpxq ds

for every m P MF7c, symbis a bijection, such that

PF6c, symb´δcpSpmqq “

PF7c, symbpmq

ş

Σ r dm.

Proof. Note thatş

Σ r dm is the total mass of the measure dµrprx, ssq “ dµpxq ds on Σr. Inparticular, Spmq is indeed a probability measure.

Since r has a positive lower bound and a finite upper bound, it is well known since [AmK],see also [IJT, §2.4], that the map S defined above is a bijection from the set of σ-invariantprobability measures m on Σ to the set of pσtrqtPR-invariant probability measures on Σr.

Furthermore, for every σ-invariant probability measure m on Σ, we have the followingKac formula, by the definition of the probability measure Spmq and by Lemma 5.15,

ż

Σr

F6c, symb dSpmq ´ δc “

ż

Σr

pF6c, symb ´ δcq dSpmq

“1

ş

Σ r dm

ż

xPΣ

ż rpxq

0pF6c, symb ´ δcqprx, ssq dmpxq ds

“1

ş

Σ r dm

ż

ΣF7c, symb dm . (5.17)

19denoted by ∆g in loc. cit.

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By the comment on the signs at the end of Lemma 5.15, this computation also proves thatthe negative part of F6c, symb is integrable for Spmq if and only if the negative part of F7c, symb

is integrable for m. Hence S is indeed a bijection from MF7c, symbto MF6c, symb

.

By Abramov’s formula [Abr], see also [IJT, Prop. 2.14], we have

hSpmqpσ1r q “

hmpσqş

Σ r dm. (5.18)

The last claim of Lemma 5.16 follows by summation from Equations (5.17) and (5.18). l

By the proof of Theorem 5.12 (replacing the potential c by 7c), the pressure of the potentialF7c, symb is equal to the critical exponent δ7c of the potential 7c, and by Lemma 5.10 (1), wehave δ7c “ 0. Hence for every m P MF7c, symb

, we have

PF6c, symb´δcpSpmqq “

PF7c, symbpmq

ş

Σ r dmďPF7c, symbş

Σ r dm“

δ7cş

Σ r dm“ 0 .

In particular, the pressure of the potential F6c, symb´ δc is at most 0, since S is a bijection. Bythe proof of Theorem 5.12 (replacing the potential c by 7c), we know that P is an equilibriumstate for the potential F7c, symb. Hence

PF6c, symb´δcpSpPqq “

PF7c, symbpPq

ş

Σ r dP“ 0 .

Therefore, SpPq is an equilibrium state of the potential F6c, symb´δc, with pressure 0. But aPr,which is equal to Pr

Pr since aPr is a probability measure, is by construction equal to SpPq.The result follows. l

With slightly different notation, this result implies Theorem 1.1 in the Introduction.

Proof of Theorem 1.1. Any potential rF for Γ on T 1X is cohomologous to a potentialrFc associated with a system of conductances (see Proposition 3.12). If two potentials rF andrF 1 for Γ on T 1X are cohomologous20 then the potentials ` ÞÑ rF pv`q and ` ÞÑ rF 1pv`q for Γ onGX are cohomologous for the definition given before the statement of Theorem 5.14. Sincethe existence and uniqueness of an equilibrium state depends only on the cohomology classof the potential on GX, the result follows. l

20See the definition at the end of Section 3.2.

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Chapter 6

Random walks on weighted graphs ofgroups

Let X be a locally finite simplicial tree without terminal vertices, and let X “ |X|1 be itsgeometric realisation. Let Γ be a nonelementary discrete subgroup of AutpXq.

In Section 6.1, we define an operator ∆c on the functions defined on the set of verticesof the quotient graph of groups ΓzzX endowed with a system of conductances c : ΓzEXÑ R.This operator is the infinitesimal generator of the random walk on ΓzzX associated with the(normalised) exponential of this system of conductances. When Γ is torsion free and thesystem of conductances vanishes, the construction recovers the standard Laplace operator onthe graph ΓzX.

Under appropriate anti-reversibility assumptions on the system of conductances, usingtechniques of Sullivan and Coornaert-Papadopoulos, we prove that the total mass of thePatterson densities is a positive eigenvector for the operator ∆c associated with the systemof conductances.

In Section 6.2, we study the nonsymmetric nearest neighbour random walks on V X associ-ated with anti-reversible systems of conductivities, and we show that the Patterson densitiesare the harmonic measures of these random walks.

6.1 Laplacian operators on weighted graphs of groups

Let X be a locally finite simplicial tree without terminal vertices, and let X “ |X|1 be itsgeometric realisation. Let Γ be a nonelementary discrete subgroup of AutpXq. Let rc : EXÑ Rbe a (Γ-invariant) system of conductances for Γ.

We define rc` “ rc and rc´ : e ÞÑ rcpeq, which is another system of conductances for Γ, andwe denote by c˘ : ΓzEXÑ R the quotient maps. Recall (see Section 3.5) that rc is reversible(respectively anti-reversible) if rc´ “ rc` (respectively rc´ “ ´rc`). For every x P V X, wedefine

degrc˘pxq “

ÿ

ePEX, opeq“xerc˘peq .

The quotient graph of groups ΓzzX is endowed with the quotient maps c˘ : ΓzEXÑ R of rc˘.Also note that the quantity deg

rc˘pxq is constant on the Γ-orbit of x. Hence, it defines a mapdegc˘ : ΓzV XÑ s0,`8r .

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On the vector space CV X of maps from V X to C, we consider the operator ∆c˘ , calledthe (weighted) Laplace operator of pX, c˘q,1 defined by setting, for all f P CV X and x P V X,

∆c˘fpxq “1

degrc˘pxq

ÿ

ePEX, opeq“xerc˘peq

`

fpxq ´ fptpeqq˘

. (6.1)

This is the standard Laplace operator2 of a weighted graph with the weight e ÞÑ erc˘peq, except

that usually one requires that rcpeq “ rcpeq. Note that p˘peq “ ec˘peq

degc˘ popeqqis a Markov transition

kernel on the tree X, see Section 6.2.The weighted Laplace operator ∆c˘ is invariant under Γ: for all f P CV X and γ P Γ, we

have∆c˘pf ˝ γq “ p∆c˘fq ˝ γ .

In particular, this operator induces an operator on functions defined on the quotient graphΓzX, as follows.

Let pY, G˚q be a graph of finite groups and let i : EY Ñ N ´ t0u be the index mapipeq “ rGopeq : Ges. For every function c : EY Ñ R, let degc : V Y Ñ R be the positivefunction defined by

degcpxq “ÿ

ePEY, opeq“xipeq ecpeq .

The Laplace operator3 of pY, G˚, cq is the operator ∆c “ ∆Y,G˚,c on L2pV Y, volY,G˚q definedby

∆cf : x ÞÑ1

degcpxq

ÿ

ePEY, opeq“xipeq ecpeq

`

fpxq ´ fptpeqq˘

.

Remark 6.1. (1) Let pY, G˚q “ ΓzzX be a graph of finite groups with p : V XÑ V Y “ ΓzV Xthe canonical projection. Let rc : EXÑ R be a potential and let c : EY “ ΓzEXÑ R be themap induced by rc. An easy computation shows that for all f P CV Y and x P V Y, we have

∆cfpxq “ ∆crfprxq

if rf “ f ˝ p : V XÑ C and rx P V X satisfies pprxq “ x.

(2) For every x P V Y, letipxq “

ÿ

ePEY, opeq“xipeq .

Then ipxq is the degree of any vertex of any universal cover of pY, G˚q above x. In particular,the map i : V Y Ñ R is bounded if and only if the universal cover of pY, G˚q has uniformlybounded degrees. When c “ 0, we denote the Laplace operator by ∆ “ ∆Y, G˚ and for everyx P V Y, we have

∆0fpxq “1

ipxq

ÿ

ePEY, opeq“xipeq

`

fpxq ´ fptpeqq˘

.

We thus recover the Laplace operator of [Mor] on the edge-indexed graph pY, iq.1or on X associated with the system of conductances rc˘2See for example [Car] with the opposite choice of the sign, or [ChGY].3See for instance [Mor] when c “ 0.

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Returning to general graphs of finite groups, we denote by L2pV Y, volpY,G˚qq the Hilbertspace of maps f : V YÑ C with finite norm fvol for the following scalar product:

xf, gyvol “ÿ

xPV Y

1

|Γx|fpxq gpxq .

We denote by L2pEY,TvolpY,G˚qq the Hilbert space of maps φ : EY Ñ C with finite normφTvol for the following scalar product:

xφ, ψyTvol “1

2

ÿ

ePEY

1

|Γe|φpeq ψpeq .

Proposition 6.2. Let pY, G˚q be a graph of finite groups, whose map i : V YÑ R is bounded.Let c : EYÑ R be a system of conductances on Y, and let

ppeq “ecpeq

degcpopeqq

for every e P EY.

(1) The Laplace operator ∆c : L2pV Y, volpY,G˚qq Ñ L2pV Y, volpY,G˚qq is linear and bounded.

(2) The map dc : L2pV Y, volpY,G˚qq Ñ L2pEY,TvolpY,G˚qq defined by

dcpfq : e ÞÑa

ppeq`

fptpeqq ´ fpopeqq˘

is linear and bounded, and its dual operator

d˚c : L2pEY,TvolpY,G˚qq Ñ L2pV Y, volpY,G˚qq

is given by

d˚c pφq : x ÞÑÿ

ePEY, opeq“x

ipeq

2

´

a

ppeq φpeq ´a

ppeq φpeq¯

(3) Assume that c is reversible. Then

∆c “ d˚cdc

In particular, ∆c is self-adjoint and nonnegative.

Proof. By the assumptions, there exists M P N such that ipxq ď M for every x P V Y, andhence ipeq ďM for every e P EY. Note that ipeq “ |Γopeq|

|Γe|, that Γe “ Γe and that ppeq ď 1 .

(1) For every f P L2pV Y, volpY,G˚qq, using the Cauchy-Schwarz inequality and the change of

93 19/12/2016

variable e ÞÑ e, we have

∆cf2vol “

ÿ

xPV Y

1

|Γx|

ˇ

ˇ

ˇ

ÿ

ePEY, opeq“xipeq ppeq

`

fpxq ´ fptpeqq˘

ˇ

ˇ

ˇ

2

ďÿ

xPV Y

1

|Γx|

´

ÿ

ePEY, opeq“xipeq2 ppeq2

¯´

ÿ

ePEY, opeq“x

ˇ

ˇfpxq ´ fptpeqqˇ

ˇ

2¯

ď 2M2ÿ

xPV Y

1

|Γx|

´

ÿ

ePEY, opeq“x

ˇ

ˇfpxqˇ

ˇ

2`ˇ

ˇfptpeqqˇ

ˇ

2¯

ď 2M2ÿ

ePEY

1

|Γe|

ˇ

ˇfpopeqqˇ

ˇ

2`ˇ

ˇfptpeqqˇ

ˇ

2

“ 4M2ÿ

ePEY

1

|Γe|

ˇ

ˇfpopeqqˇ

ˇ

2“ 4M2

ÿ

xPV Y

1

|Γx|

ÿ

ePEY, opeq“xipeq |fpxq|2

ď 4M3ÿ

xPV Y

1

|Γx||fpxq|2 “ 4M3 f2vol .

Hence the linear operator ∆c is bounded.

(2) For every f P L2pV Y, volpY,G˚qq, we have

dcf2Tvol “

1

2

ÿ

ePEY

ppeq

|Γe|

ˇ

ˇfptpeqq ´ fpopeqqˇ

ˇ

2

ďÿ

ePEY

1

|Γe|

´

ˇ

ˇfptpeqqˇ

ˇ

2`ˇ

ˇfpopeqqˇ

ˇ

2¯

“ 2ÿ

ePEY

1

|Γe|

ˇ

ˇfpopeqqˇ

ˇ

2

“ 2ÿ

ePEY

ipeq

|Γopeq|

ˇ

ˇfpopeqqˇ

ˇ

2“ 2

ÿ

xPV Y

ipxq

|Γx||fpxq|2 ď 2Mf2vol .

Hence the linear operator dc is bounded.For all f P L2pV Y, volpY,G˚qq and φ P L2pEY,TvolpY,G˚qq, using again the change of

variable e ÞÑ e, we have

xφ, dcfyTvol “1

2

ÿ

ePEY

1

|Γe|

a

ppeq φpeq`

fptpeqq ´ fpopeq˘

“1

2

ˆ

ÿ

ePEY

a

ppeq

|Γe|φpeq fpopeqq ´

ÿ

ePEY

a

ppeq

|Γe|φpeq fpopeqq

˙

“ÿ

xPV Y

1

|Γx|

ÿ

ePEY, opeq“x

ipeq

2

´

a

ppeq φpeq ´a

ppeq φpeq¯

fpxq .

This gives the formula for d˚c .

(3) For all f, g P L2pV Y, volpY,G˚qq, since ppeq “ ppeq by reversibility, by making the change

94 19/12/2016

of variable e ÞÑ e in half the value of the second line, we have

x∆cf, gyvol “ÿ

xPV Y

1

|Γx|

ÿ

ePEY, opeq“xipeq ppeq

`

fpxq ´ fptpeqq˘

gpxq

“ÿ

ePEY

ipeq

|Γopeq|ppeq

`

fpopeqq gpopeqq ´ fptpeqq gpopeqq˘

“1

2

ÿ

ePEY

1

|Γe|ppeq

`

fptpeqq ´ fpopeqq˘`

gptpeqq ´ gpopeqq˘

“ xdcf, dcgyTvol .

This proves the last claim in Proposition 6.2. l

The following result is an extension to anti-reversible systems of conductances of [CoP2,Prop. 3.3] (who treated the case of zero conductances), which is a discrete version of Sullivan’sanalogous result for hyperbolic manifolds (see [Sul1]). Let rFc : T 1X Ñ R be the potentialfor Γ associated with c, so that p rFcq˘ “ rFc˘ , and let δc be their critical exponent. LetC˘ : B8XˆV XˆV XÑ R be the associated Gibbs cocycles. Let pµ˘x qxPV X be two Pattersondensities on B8X for the pairs pΓ, Fc˘q.

Proposition 6.3. Assume that X is pq ` 1q-regular, that the system of conductances c isanti-reversible and that the map degc˘ : V X Ñ R is constant with value κ˘. Then the totalmass rφ˘ : x ÞÑ µ˘x of the Patterson density is a positive eigenvector associated with theeigenvalue

1´eδc ` qe´δc

κ˘.

for the Laplace operator ∆c˘ on CV X.

Proof. Note that c˘ : V X Ñ R is bounded, hence c˘ : EX Ñ R is bounded, hencep rFcq

˘ “ rFc˘ is bounded. As X is pq ` 1q-regular, the critical exponent δΓ is finite and hencealso the critical exponent δc is finite by Assertions (6) and (7) of Lemma 3.7. Since

rφ˘pxq “

ż

B8Xdµ˘x “

ż

B8Xe´C

˘ξ px, x0q dµ˘x0

,

by Equation (4.2) and by linearity, we only have to prove that for every fixed ξ P B8X themap

f : x ÞÑ e´C˘ξ px, x0q

is an eigenvector with eigenvalue 1´ eδc`qe´δc

κ˘for ∆c˘ .

For every e P EX, recall4 that BeX is the set of points at infinity of the geodesic rays whoseinitial edge is e. By Equation (3.8) and by the definition of the potential associated with asystem of conductances5, for all e P EX and η P BeX, we have

C˘η ptpeq, opeqq “

ż tpeq

opeqp rFc˘ ´ δcq “ c˘peq ´ δc .

4See Section 2.7.5See Proposition 3.11 with the edge length map λ constant equal to 1

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Thusfptpeqq “ e´C

˘ξ ptpeq, opeqq´C

˘ξ popeq, x0q “ e´c

˘peq`δc fpopeqq

if ξ P BeX, and otherwise

fptpeqq “ eC˘ξ ptpeq, opeqq´C

˘ξ popeq, x0q “ ec

˘peq´δc fpopeqq .

For every x P V X, let eξ be the unique edge of X with origin x such that ξ P BeξX. Then,

∆c˘fpxq “ fpxq ´1

c˘pxq

ÿ

opeq“x

ec˘peqfptpeqq

“ fpxq ´1

κ˘ec˘peξqfptpeξqq ´

1

κ˘

ÿ

e‰eξ, opeq“x

ec˘peqfptpeqq

“

´

1´eδc

κ˘´q e´δc

κ˘

¯

fpxq .

This proves the result. l

Note that the anti-reversibility of the potential is used in an essential way in order to getthe last equation in the proof of Proposition 6.3.

6.2 Patterson densities as harmonic measures for simplicialtrees

In this Section, we define and study a Markov chain on the set of vertices of a simplicialtree endowed with a discrete group of automorphisms and with an appropriate system ofconductances, such that the associated (nonsymmetric, nearest neighbour) random walk con-verges almost surely to points in the boundary of the tree, and we prove that the Pattersondensities, once normalised, are the corresponding harmonic measures. We thereby gener-alise the zero potential case treated in [CoP2], which is also a special case of [CoM] whenX is a tree under the additional restriction that the discrete group is cocompact. For otherconnections between harmonic measures and Patterson measures, we refer for instance to[CoM, BlHM, Tan, GouMM] and their references.

Let X be a pq ` 1q-regular simplicial tree, with q ě 2. Let Γ be a nonelementary discretesubgroup of AutpXq. Let rc : EX Ñ R be an anti-reversible system of conductances for Γ,such that the associated map rc : V X Ñ R on the vertices of X is constant. Let pµxqxPV X bea Patterson density for pΓ, Fcq, where Fc is the potential associated with c. We denote byφµ : x ÞÑ µx the associated total mass function on V X.

We start this Section by recalling a few facts about discrete Markov chains, for which werefer for instance to [Rev, Woe1]. A state space is a discrete and countable set I. A transitionkernel on a I is a map P : I ˆ I Ñ r0, 1s such that for every x P I,

ÿ

yPI

P px, yq “ 1 .

Let λ be a probability measure on I. A (discrete) Markov chain on a state space I with initialdistribution λ and transition kernel P is a sequence pZnqnPN of random variables with valuesin I such that for all n P N and x0, . . . , xn`1 P I, the probability of events P satisfies

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(1) PrZ0 “ x0s “ λptx0uq,

(2) PrZn`1 “ xn`1 | Z0 “ x0, Z1 “ x1, . . . , Zn “ xns “ P pxn, xn`1q.

The associated random walk consists in choosing a point x0 in I with law λ, and by induction,once xn is constructed, in choosing xn`1 in I with probability P pxn, xn`1q. Note that

PrZ0 “ x0, Z1 “ x1, . . . , Zn “ xns “ λptx0uqP px0, x1q . . . P pxn´1, xnq .

When the initial distribution λ is the unit Dirac mass ∆x at x P I, the Markov chain is thenuniquely determined by its transition kernel P and by x, and is denoted by pZxnqnPN.

For every n P N, we denote by P pnq the iterated matrix product of the transition kernel P :we have P p1q “ P , P p0qpx, yq is the Kronecker symbol δx,y for all x, y P I, and by inductionP pn`1q “ P ¨ P pnq where ¨ is the matrix product of I ˆ I matrices, that is, for all x, z P I,

P pn`1qpx, zq “ÿ

yPI

P px, yqP pnqpy, zq .

Note thatP pnqpx, yq “ PrZxn “ ys

is the probability for the random walk starting at time 0 from x of being at time n at thepoint y. The Green kernel of P is the map GP from I ˆ I to r0,`8s defined by

px, yq ÞÑ GP px, yq “ÿ

nPNP pnqpx, yq ,

and its Green function is the following power series in the complex variable z :

GP px, y | zq “ÿ

nPNP pnqpx, yq zn .

Recall that if GP px, yq ‰ 0 for all x, y P I, then the random walk is recurrent6 if GP px, yq “ 8for any (hence all) px, yq P I ˆ I, and transient otherwise. Note that, using again matrixproducts of I ˆ I matrices,

GP “ Id`P ¨GP . (6.2)

We will from now on consider as state space the set V X of vertices of X. If a Markov chainpZxnqnPN starting at time 0 from x converges almost surely in V XYB8X to a random variableZx8, the law of Zx8 is called the harmonic measure (or hitting measure on the boundary)associated with this Markov chain, and is denoted by

νx “ pZx8q˚pPq .

Note that νx is a probability measure on B8X.For instance, the transition kernel of the simple nearest neighbour random walk on X is

defined by taking as transition kernel the map P where

P px, yq “1

q ` 1Apx, yq

6that is, Cardtn P N : Zxn “ yu “ 8 for every y P I (or equivalently, there exists y P I such thatCardtn P N : Zxn “ yu “ 8)

97 19/12/2016

for all x, y P V X, with A : V Xˆ V XÑ t0, 1u the adjacency matrix of the tree X, defined byApx, yq “ 1 for any two vertices x, y of X that are joined by an edge in X and Apx, yq “ 0otherwise. We denote by

Gpx, y | zq “ÿ

kPNP pnqpx, yq zn .

the Green function of P , whose radius of convergence is r “ q`12?q and which diverges at z “ r,

see for example [Woe1], [Woe2, Ex. 9.82], [LyP2, §6.3].

The antireversible system of conductances rc : EX Ñ R defines a cocycle on the set ofvertices of X, as follows. For every u, v in V X, let cpu, vq “ 0 if u “ v and otherwise let

cpx, yq “nÿ

i“1

rcpeiq ,

where pe1, e2, . . . , emq is the geodesic edge path in X from u “ ope1q to v “ tpenq.

Lemma 6.4. (1) For every edge path pe11, e12, . . . , e

1n1q from u to v, we have

cpu, vq “n1ÿ

i“1

rcpe1iq .

(2) The map c : V Xˆ V XÑ R has the following cocycle property: for all u, v, w P V X,

cpu, vq ` cpv, wq “ cpu,wq and hence cpv, uq “ ´cpu, vq .

(3) We have cpu, vq “şvurFc.

(4) For all ξ P B8X and u, v P V X, if Cc¨ p¨, ¨q is the Gibbs cocycle associated with rFc, wehave

Ccξpu, vq “ cpv, uq ` δcβξpu, vq .

Proof. (1) Since X is a simplicial tree, any nongeodesic edge path from u to v has a back-and-forth on some edge, which contributes to 0 to the sum defining cpx, yq by the anti-reversibilityassumption on the system of conductances. Therefore, by induction, the sum in Assertion (1)indeed does not depend on the choice of the edge path from u to v.

Assertion (2) is immediate from Assertion (1). Assertion (3) follows from the definition ofcp¨, ¨q by Proposition 3.11.

(4) For every ξ P B8X, if p P V X is such that ru, ξr X rv, ξr “ rp, ξr , then using Equation (3.6)and Assertions (3) and (2), we have

Ccξpu, vq “

ż p

vp rFc ´ δcq ´

ż p

up rFc ´ δcq “ cpv, pq ´ cpu, pq ` δc βξpu, vq

“ cpv, uq ` δc βξpu, vq . l

We now define the transition kernel Pc associated with7 the (logarithmic) system of con-ductances c by, for all x, y P V X,

Pcpx, yq “ κcφµpyq

φµpxqecpx, yq P px, yq .

7The transition kernel also depends on the choice of the Patterson density if Γ is not of divergence type.

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From now on, we denote by pZxnqnPN the Markov chain with initial distribution ∆x andtransition kernel Pc.

Letκc “

1` q

eδc ` q e´δc,

which belongs to s0, q`12?q s . Note that this constant κc is less than the radius of convergence

r of the Green function Gpx, y | zq if and only if coshpδc ´12 ln qq ą 1, that is, if and only

if δc ‰ 12 ln q. The computation (due to Kesten) of the Green function of P is well known,

and gives the following formula, see for instance [CoP2, Prop. 3.1]: If δc ‰ 12 ln q, then there

exists α ą 0 such that for all x, y P V X

Gpx, y | κcq “ α e´δc dpx, yq . (6.3)

Lemma 6.5. (1) The map Pc is indeed a transition kernel on V X.

(2) The Green kernel Gc “ GPc of Pc is

Gcpx, yq “ ecpx, yqφµpyq

φµpxqGpx, y | κcq . (6.4)

In particular, the Green kernel of Pc is finite if δc ‰ 12 ln q.

(3) Assume that δc ‰ 12 ln q. For all x, y, z P V X, we have

φµpyq Gcpy, zq

φµpxq Gcpx, zq“ ecpy, xq`δcpdpx, zq´dpy, zqq .

If furthermore z R rx, yr , then, for every ξ P Oxpzq,

φµpyq Gcpy, zq

φµpxq Gcpx, zq“ eC

cξpx, yq .

Proof. (1) By Proposition 6.3, the positive function φµ is an eigenvector with eigenvalueeδc ` q e´δc for the operator

f ÞÑ tx ÞÑÿ

ePEX, opeq“xercpeq fptpeqqu .

Since P popeq, tpeqq “ 1q`1 for every e P EX, we hence have

ÿ

yPV XPcpx, yq “

ÿ

ePEX, opeq“xPcpx, tpeqq

“1` q

peδc ` q e´δcq φµpxq

ÿ

ePEX, opeq“xercpeq φµptpeqq P px, tpeqq “ 1 .

(2) Let us first prove that for all x, y P V X and n P N, we have

P pnqc px, yq “ pκcqn φµpyq

φµpxqecpx, yq P pnqpx, yq . (6.5)

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Indeed, by the cocycle property of cp¨, ¨q and by a telescopic cancellation argument, we have

P pnqc px, yq

“ÿ

x1,..., xn´1PV XPcpx, x1q Pcpx1, x2q . . . Pcpxn´2, xn´1q Pcpxn´1, yq

“ pκcqn φµpyq

φµpxqecpx, yq

ÿ

x1,..., xn´1PV XP px, x1q P px1, x2q . . . P pxn´2, xn´1q P pxn´1, yq

“ pκcqn φµpyq

φµpxqecpx, yq P pnqpx, yq .

Equation (6.4) follows from Equation (6.5) by summation on n. As we have already seen,κc ă r if and only if δc ‰ 1

2 ln q. The last claim of Assertion (2) follows.

(3) Let x, y, z P V X. Using (twice) Assertion (2), the cocycle property of c and (twice)Equation (6.3), we have

Gcpy, zq

Gcpx, zq“ecpy, zq φµpzq φµpxq Gpy, z | κcq

ecpx, zq φµpyq φµpzq Gpx, z | κcq“ ecpy, xq

φµpxq

φµpyq

α e´δc dpy, zq

α e´δc dpx, zq

“φµpxq

φµpyqecpy, xq`δcpdpx, zq´dpy, zqq .

This proves the first claim of Assertion (3). Under the additional assumptions on x, y, z, ξ,we have

βξpx, yq “ dpx, zq ´ dpy, zq .

The last claim of Assertion (3) hence follows from Lemma 6.4 (4). l

Using the criterion that the random walk starting from a given vertex of X with transitionprobabilities Pc is transient if and only if the Green kernel Gcpx, yq of Pc is finite (for any,hence for all, x, y P X), Lemma 6.5 (2) implies that if δc ‰ 1

2 ln q, then pZxnqnPN almost surelyleaves every finite subset of V X. The following result strengthens this remark.

Proposition 6.6. If δc ‰ 12 ln q, then for every x P V X, the Markov chain pZxnqnPN (with

initial distribution ∆x and transition kernel Pc) converges almost surely in V X Y B8X to arandom variable with values in B8X. In particular the harmonic measure νx of pZxnqnPN iswell defined if δc ‰ 1

2 ln q.

Proof. Since X is a tree, if pxnqnPN is a sequence in V X such that dpxn, xn`1q “ 1 for everyn P N and which does not converge to a point in B8X, then there exists a point y such thatthis sequence passes infinitely often through y, that is, tn P N : xn “ yu is infinite. Theresult then follows from the fact that the Markov chain pZxnqnPN is transient since δc ‰ 1

2 ln q.l

The following result, generalising [CoP2] when c “ 0, says that the Patterson measuresassociated with a system of conductances c, once renormalised to probability measures, areexactly the harmonic measures for the random walk with transition probabilities Pc.

Theorem 6.7. Let pX,Γ,rc, pµxqxPV Xq be as in the beginning of Section 6.2. If δc ‰ 12 ln q,

then for every x P X, the harmonic measure of the Markov chain pZxnqnPN is

νx “µxµx

.

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Proof. We fix x P X. Recall that given z P V X, the shadow Oxpzq of z seen from x is the setof points at infinity of the geodesic rays from x through z.

For every n P N, we denote by Spx, nq and Bpx, nq the sphere and ball of centre x andradius n in V X, and we define two maps f1, f2 : V XÑ R with finite support by

f1pzq “µxpOxpzqq

µx Gcpx, zqand f2pzq “

νxpOxpzqq

Gcpx, zq

if z P Spx, nq, and f1pzq “ f2pzq “ 0 otherwise. Let us prove that f1 “ f2 for every n P N.Since tOxpzq : z P V Xu generates the Borel σ-algebra of B8X, this proves that the Borelmeasures νx and µx

µxcoincide.

We will use the following criterion: For all maps G : V XˆV XÑ R and f : V XÑ R suchthat f has finite support, let us again denote by G ¨ f the matrix product of G and f : forevery y P V X,

G ¨ fpyq “ÿ

zPV XGpy, zqfpzq .

Lemma 6.8. For all f, f 1 : V XÑ R with finite support, if Gc ¨ f “ Gc ¨ f1, then f “ f 1.

Proof. By Equation (6.2), we have

f 1 “ Gc ¨ f1 ´ Pc ¨Gc ¨ f

1 “ Gc ¨ f ´ Pc ¨Gc ¨ f “ f . l

Let us hence fix n P N and prove that Gc ¨ f1 “ Gc ¨ f2. Theorem 6.7 then follows.

Step 1: For every y P Bpx, nq, since tOxpzq : z P Spx, nqu is a Borel partition of B8X,by Equation (4.2), since z R rx, yr if z P Spx, nq and y P Bpx, nq, and by the second claim ofLemma 6.5 (3), we have

1 “1

φµpyq

ż

B8Xdµy “

1

φµpyq

ÿ

zPSpx,nq

ż

Oxpzqe´C

cξpy, xq dµxpξq

“1

φµpyq

ÿ

zPSpx,nq

ż

Oxpzq

φµpyq Gcpy, zq

φµpxq Gcpx, zqdµxpξq

“ÿ

zPSpx,nq

Gcpy, zqµxpOxpzqq

µx Gcpx, zq“ Gc ¨ f1pyq . (6.6)

Step 2: For all y, z P V X such that z R rx, yr , any random walk starting at time 0 from yand converging to a point in Oxpzq goes through z. Let us denote by Cxpzq the set of verticesdifferent from z on the geodesic rays from z to the points in Oxpzq. Partioning by the lasttime the random walk passes through z, using the Markov property saying that what happensbefore the random walk arrives at z and after it leaves z are independent, we have

νypOxpzqq “ PrZy8 P Oxpzqs “ Gcpy, zq Pr@n ą 0, Zzn P Cxpzqs ,

so thatνypOxpzqq

νxpOxpzqq“

Gcpy, zq

Gcpx, zq. (6.7)

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Step 3: For every y P Bpx, nq, again since tOxpzq : z P Spx, nqu is a Borel partition ofB8X, and by Equation (6.7), we have

1 “ νy “ÿ

zPSpx,nq

νypOxpzqq “ÿ

zPSpx,nq

Gcpy, zqνxpOxpzqq

Gcpx, zq“ Gc ¨ f2pyq . (6.8)

Step 4: By Steps 1 and 3, we have Gc ¨ f1pyq “ Gc ¨ f2pyq for every y P Bpx, nq. Let nowy P V X ´ Bpx, nq. Since x P Bpx, nq, we have as just said Gc ¨ f1pxq “ Gc ¨ f2pxq. Hence bythe first claim of Lemma 6.5 (3), we have

Gc ¨ f1pyq “ÿ

zPSpx,nq

Gcpy, zq f1pzq

“ÿ

zPSpx,nq

ecpy, xq`δcpdpx, zq´dpy, zqqφµpxq

φµpyqGcpx, zq f1pzq

“ ecpy, xq`δcpdpx, zq´dpy, zqqφµpxq

φµpyqGc ¨ f1pxq

“ ecpy, xq`δcpdpx, zq´dpy, zqqφµpxq

φµpyqGc ¨ f2pxq “ Gc ¨ f2pyq .

This proves that Gc ¨ f1 “ Gc ¨ f2, thereby concluding the proof of Theorem 6.7. l

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Chapter 7

Skinning measures with potential onCATp´1q spaces

In this Chapter, we introduce skinning measures as weighted pushforwards of the Patterson-Sullivan densities associated with a potential to the unit normal bundles of convex subsets ofa CATp´1q space. The development follows [PaP14a] with modifications to fit the presentcontext.

Let X,x0,Γ, rF be as in the beginning of Chapter 4. Let pµ˘x qxPX be Patterson densitieson B8X for the pairs pΓ, F˘q.

7.1 Skinning measures

Let D be a nonempty proper closed convex subset of X. The outer skinning measure rσ`Don B1

`D and the inner skinning measure rσ´D on B1´D associated with the Patterson densities

pµ˘x qxPX for pΓ, rF˘q are the measures rσ˘D “ rσ˘D,F˘

defined by

drσ˘Dpρq “ eC˘ρ˘px0, ρp0qq dµ˘x0

pρ˘q , (7.1)

where ρ P B1˘D, using the endpoint homeomorphisms ρ ÞÑ ρ˘ from B1

˘D to B8X ´ B8D, andnoting that ρp0q “ PDpρ˘q depends continuously on ρ˘.

When rF “ 0, the skinning measure has been defined by Oh and Shah [OhS2] for theouter unit normal bundles of spheres, horospheres and totally geodesic subspaces in real hy-perbolic spaces. The definition was generalised in [PaP14a] to the outer unit normal bundlesof nonempty proper closed convex sets in Riemannian manifolds with variable negative cur-vature.

Note that the Gibbs measure is defined on the space GX of geodesic lines, the potentialis defined on the space T 1X of germs at time t “ 0 of geodesic lines, and since B1

˘D iscontained in G˘, 0X (see Section 2.5), the skinning measures are defined on the spaces G˘, 0X of(generalised) geodesic rays. In the manifold case, all the above spaces are canonically identifiedwith the unit tangent bundle, but in general, the natural restriction maps GX Ñ T 1X andGX Ñ G˘, 0X have infinite (though compact) fibers.

Remark 7.1. (1) If D “ txu is a singleton, then

drσ˘Dpρq “ dµ˘x pρ˘q (7.2)

103 19/12/2016

where ρ is a geodesic ray starting (at time t “ 0) from x.

(2) When the potential rF is reversible (in particular when F “ 0), we have C´ “ C`, wemay (and we will) take µ´x “ µ`x for all x P X, hence ι˚ rmF “ rmF and rσ´D “ ι˚rσ

`D.

(3) The (normalised) Gibbs cocycle being unchanged when the potential F is replaced by thepotential F `σ for any constant σ, we may (and will) take the Patterson densities, hence theGibbs measure and the skinning measures, to be unchanged by such a replacement.

When D is a horoball in X, let us now related the skinning measures of D with previouslyknown measures on B8X, constructed using techniques due to Hamenstädt.

Let H be a horoball centred at a point ξ P B8X. Recall that PH : B8X ´ tξu Ñ BH isthe closest point map on H , mapping η ‰ ξ to the intersection with the boundary of H ofthe geodesic line from η to ξ. The following result is proved in [HeP3, §2.3] when F “ 0.

Proposition 7.2. Let ρ : r0,`8rÑ X be the geodesic ray starting from any point of theboundary of H and converging to ξ. The following weak-star limit of measures on B8X´tξu

dµ˘H pηq “ limtÑ`8

e´şPH pηq

ρptqp rF˘´δq

dµ˘ρptqpηq

exists, and it does not depend on the choice of ρ. The measure µ˘H is invariant under theelements of Γ preserving H , and it satisfies, for every x P X and (almost) every η P B8X ´tξu,

dµ˘Hdµ˘x

pηq “ e´C˘η pPH pηq, xq .

Proof. We prove all three assertions simultaneously. Let us fix x P X. For all t ě 0 andη P B8X ´ tξu, let zt be the closest point to PH pηq on the geodesic ray from ρpyq to η.

ηξ

ρp0q

zt

pH pηq

ρptq

Using Equation (4.2) with x replaced by ρptq and y by the present x, by the cocycleequation (3.7) and by Equation (3.8) as zt P rρptq, ηr , we have

e´şPH pηq

ρptqp rF˘´δq

dµ˘ρptqpηq “ e´şPH pηq

ρptqp rF˘´δq

e´C˘η pρptq, xq dµ˘x pηq

“ e´şPH pηq

ρptqp rF˘´δq

e´C˘η pρptq, ztq e´C

˘η pzt, xq dµ˘x pηq

“ e´şPH pηq

ρptqp rF˘´δq`

şztρptqp rF˘´δq

e´C˘η pzt, xq dµ˘x pηq .

As t Ñ `8, note that zt converges to PH pηq and that by the HC-property (and since rF isbounded on any compact neighbourhood of PH pηq), we have

ˇ

ˇ

ˇ

ż PH pηq

ρptqp rF˘ ´ δq ´

ż zt

ρptqp rF˘ ´ δq

ˇ

ˇ

ˇÑ 0 .

104 19/12/2016

The result then follows by the continuity of the Gibbs cocycle (see Proposition 3.10 (3)). l

Using this proposition and the cocycle property of C˘ in the definition (4.3) of the Gibbsmeasure, we obtain, for every ` P GX such that `˘ ‰ ξ,

drmF p`q “ eC´`´

pPH p`´q, `p0qq`C```pPH p``q, `p0qq

dµ´H p`´q dµ`H p``q dt . (7.3)

Note that it is easy to see that for every ρ P B1˘H , we have

drσ˘H pρq “ dµ˘H pρ˘q . (7.4)

When F “ 0, we obtain Hamenstädt’s measure

µH “ limtÑ`8

eδΓ tµρptq (7.5)

on B8X ´ tξu associated with the horoball H , which is independent of the choice of thegeodesic ray ρ starting from a point of the horosphere BH and converging to ξ. Note thatfor every t ě 0, if H rts is the horoball contained in H whose boundary is at distance t fromthe boundary of H , we then have

µH rts “ e´δΓ t µH . (7.6)

The following results give the basic properties of the skinning measures analogous to thosein [PaP14a, Sect. 3] when the potential is zero.

Proposition 7.3. Let D be a nonempty proper closed convex subset of X, and let rσ˘D be theskinning measures on B1

˘D for the potential rF .(i) The skinning measures rσ˘D are independent of x0.(ii) For all γ P Γ, we have γ˚rσ˘D “ rσ˘γD. In particular, the measures rσ˘D are invariant underthe stabiliser of D in Γ.(iii) For all s ě 0 and w P B1

˘D, denoting by pg˘swq|˘r0,`8r the element of G˘, 0 whichcoincides with g˘sw on ˘r0,`8r, we have

d rσ˘NsDppg˘swq|˘r0,`8rq “ eC

˘w˘pπpwq, πpg˘swqq drσ˘Dpwq “ e

´şπpg˘swqπpwq

p rF˘´δqdrσ˘Dpwq .

(iv) The support of rσ˘D is

tv P B1˘D : v˘ P ΛΓu “ P˘D pΛΓ´ pΛΓX B8Dqq .

In particular, rσ˘D is the zero measure if and only if ΛΓ is contained in B8D.

For future use, the version1 of Assertion (iii) when F “ 0 is

dpg˘sq˚rσ˘D

d rσ˘NsD

pg˘swq “ e´δ s . (7.7)

As another particular case of Assertion (iii) for future use, consider the case whenX “ |X|λis the geometric realisation of a metric tree pX, λq and when rF “ rFc is the potential associatedwith a system of conductances c on X (see Equation (3.11) and Proposition 3.12). Then for

1contained in [PaP14a, Prop. 4]

105 19/12/2016

all w P B1`D (respectively w P B1

´D), if ew is the first (respectively the last) edge followed byw, with length λpeωq, then

ż πpg˘λpeωqwq

πpwq

rF˘ “ cpewq

by Proposition 3.11, so that

d rσ˘NsDppg˘λpeωqwq|˘r0,`8rq “ e´cpeωq`δλpeωq drσ˘Dpwq . (7.8)

Proof. We give details only for the proof of claim (iii) for the measure rσ`D, the case ofrσ´D being similar, and the proofs of the other claims being straightforward modifications ofthose in [PaP14a, Prop. 4]. Since

`

pgswq|r0,`8r˘

`“ w` and since w P B1

`D if and only ifpgswq|r0,`8r P B

1`NsD, we have, using the definition of the skinning measure and the cocycle

property (3.7), for all s ě 0,

d rσ`NsDppgswq|r0,`8rq “ eC

`w`px0, πpgswqq dµ`x0

pw`q “ eC`w`pπpwq, πpgswqq d rσ`Dpwq .

This proves the claim (iii) for rσ`D, using Equation (3.8). l

Given two nonempty closed convex subsets D and D1 of X, let

AD,D1 “ B8X ´ pB8D Y B8D1q

and let h˘D,D1 : P˘D pAD,D1q Ñ P˘D1pAD,D1q be the restriction of P˘D1 ˝ pP˘D q´1 to P˘D pAD,D1q. It

is a homeomorphism between open subsets of B1˘D and B1

˘D1, associating to the element w

in the domain the unique element w1 in the range with w1˘ “ w˘. The proof of Proposition5 of [PaP14a] generalises immediately to give the following result.

Proposition 7.4. Let D and D1 be nonempty closed convex subsets of X and h˘ “ h˘D,D1 .The measures ph˘q˚ rσ˘D and rσ˘D1 on P

˘D1pAD,D1q are absolutely continuous one with respect to

the other, withdph˘q˚ rσ

˘D

drσ˘D1pw1q “ e´C

˘w˘pπpwq, πpw1qq,

for (almost) all w P P˘D pAD,D1q and w1 “ h˘pwq. l

Let w P G˘X. With N˘w : W˘pwq Ñ B1¯HB˘pwq the canonical homeomorphism defined in

Section 2.5, we define the skinning measures µW˘pwq on the strong stable or strong unstableleaves W˘pwq by

µW˘pwq “ ppN˘w q´1q˚rσ

¯

HB˘pwq,

so thatdµW˘pwqp`q “ e

C¯`¯px0, `p0qq

dµ¯x0p`¯q (7.9)

for every ` P W˘pwq. By Proposition 7.3 (ii) and the naturality of N˘w , for every γ P Γ, wehave

γ˚µW˘pwq “ µW˘pγwq . (7.10)

By Proposition 7.3 (iv), the support of µW˘pwq is t` PW˘pwq : `¯ P ΛΓu. For all t P R and` PW˘pwq, we have, using Equations (3.7) and (3.8), and since `˘ “ w˘,

dpg´tq˚µW˘pwq

dµW˘pgtwqpgt`q “ e

C¯`¯p`ptq, `p0qq

“ eC˘w˘p`p0q, `ptqq . (7.11)

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Let w P G˘X. The homeomorphisms W˘pwq ˆRÑW 0˘pwq, defined by setting p`, sq ÞÑ`1 “ gs`, conjugate the actions of R by translation on the second factor of the domain and bythe geodesic flow on the range, and the actions of Γ (trivial on the second factor of the domain).Let us consider the measures ν¯w on W 0˘pwq given, using the above homeomorphism, by

dν¯w p`1q “ eC

˘w˘pwp0q, `p0qq dµW˘pwqp`q ds . (7.12)

They satisfy pgtq˚ν˘w “ ν˘w for all t P R (since if `1 “ gs`, then g´t`1 “ gs´t`). Furthermore,γ˚ν

˘w “ ν˘γw for all γ P Γ. In general, they depend on w, not only on W˘pwq. Furthermore,

the support of ν˘w is t`1 P W 0˘pwq : `1¯ P ΛΓu. These properties follow easily from theproperties of the skinning measures on the strong stable or strong unstable leaves.

Lemma 7.5. (i) For every nonempty proper closed convex subset D1 in X, there exists R0 ą 0such that for all R ě R0, η ą 0, and w P B1

˘D1, we have ν¯w pV

˘w, η,Rq ą 0.

(ii) For all w P G˘X and t P R, the measures ν¯gtw and ν¯w are proportional:

ν¯gtw “ eC

˘w˘pwptq, wp0qq ν¯w .

Proof. (i) By [PaP14a, Lem. 7],2 there exists R0 ą 0 (depending only on D1 and onthe Patterson densities) such that for all R ě R0, w P B1

`D1 and w1 P B1

´D1, we have

µW`pwqpB`pw,Rqq ą 0 and µW´pw1qpB

´pw1, Rqq ą 0. The result hence follows by the defini-tions of ν¯w and V ˘w, η,R.

(ii) For all w P G˘X, s, t P R and ` P W˘pwq, we have by Equations (7.12) and (7.11), andby the cocycle property of C˘,

dν¯gtwpg

s`q “ dν¯gtwpg

s´tgt`q “ eC˘pgtwq˘

pgtwp0q, gt`p0qqdµW˘pgtwqpg

t`q dps´ tq

“ eC˘w˘pwptq, `ptqq e´C

˘w˘p`p0q, `ptqq dµW˘pwqp`q ds

“ eC˘w˘pwptq, `p0qq e´C

˘w˘pwp0q, `p0qq dν¯w pg

s`q

“ eC˘w˘pwptq, wp0qqdν¯w pg

s`q . l

The following disintegration result of the Gibbs measure over the skinning measures ofany closed convex subset is a crucial tool for our equidistribution and counting results. Recallthe definition in Equation (2.10) of the flow-invariant open sets U ˘

D and the definition of thefibrations f˘D : U ˘

D Ñ B1˘D from Section 2.5.

Proposition 7.6. Let D be a nonempty proper closed convex subset of X. The restrictionto U ˘

D of the Gibbs measure rmF disintegrates by the fibration f˘D : U ˘D Ñ B1

˘D, over theskinning measure rσ˘D of D, with conditional measure ν¯ρ on the fiber pf˘D q

´1pρq “W 0˘pρq ofρ P B1

˘D: when ` ranges over U ˘D , we have

drmF |U ˘Dp`q “

ż

ρPB1˘D

dν¯ρ p`q drσ˘Dpρq .

Proof. In order to prove the claim for the fibration f`D , let φ P CcpU`D q. Using in the various

steps below:2whose proof extends to the present situation, although the notation is different.

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• Hopf’s parametrisation with time parameter t and the definitions of rmF (see Equation(4.3)) and of U `

D (see Equation (2.10)),• the positive endpoint homeomorphism w ÞÑ w` from B1

`D to B8X´B8D, and the negativeendpoint homeomorphism v1 ÞÑ v1´ from W`pwq to B8X ´ tw`u, with s P R the realparameter such that v1 “ g´sv P W`pwq where v P W 0`pwq, noting that t ´ s dependsonly on v` “ w` and v´ “ v1´,

• the definitions of the measures µW`pwq (see Equation (7.9)) and rσ`D (see Equation (7.1))and the cocycle property of C˘,

• Equation (3.8) and the cocycle properties of C`,we have

ż

vPU `D

φpvq drmF pvq

“

ż

v`PB8X´B8D

ż

v´PB8X´tv`u

ż

tPRφpvq e

C´v´px0, πpvqq`C

`v`px0, πpvqq dt dµ´x0

pv´q dµ`x0pv`q

“

ż

wPB1`D

ż

v1PW`pwq

ż

sPRφpgsv1q e

C´v1´

px0, πpgsv1qq`C`w`

px0, πpgsv1qq

ds dµ´x0pv1´q dµ

`x0pw`q

“

ż

wPB1`D

ż

v1PW`pwq

ż

sPRφpgsv1q e

C´v1´

pπpv1q, πpgsv1qq`C`w`pπpwq, πpgsv1qq

ds dµW`pwqpv1q drσ`Dpwq

“

ż

wPB1`D

ż

v1PW`pwq

ż

sPRφpgsv1q e

C`w`pπpwq, πpv1qq

ds dµW`pwqpv1q drσ`Dpwq ,

which implies the claim for the fibration f`D . The proof for the fibration f´D is similar. l

In particular, for every u P G´X, applying the above proposition and a change of variableto D “ HB´puq for which B1

`D “ N´u pW´puqq and

U `D “ GX ´W 0`pιuq “

ď

wPW´puq

W 0`pwq ,

the restriction to GX ´ W 0`pιuq of the Gibbs measure rmF disintegrates over the strongunstable measure µW´puq “ ppN

´u q´1q˚rσ

`D, with conditional measure on the fiber W 0`pwq of

w PW´puq the measure ν´w “ ν´N´u pwq

: for every φ P CcpGX ´W 0`pιuqq, we have

ż

GX´W 0`pιuqφpvq drmF pvq “

ż

wPW´puq

ż

v1PW`pwq

ż

sPRφpgsv1q eC

`w`pπpwq, πpv1qq ds dµ´

W`pwqpv1q dµW´puqpwq . (7.13)

Note that if the Patterson densities have no atoms, then the stable and unstable leaves havemeasure zero for the associated Gibbs measure. This happens for instance if the Gibbs measuremF is finite, see Corollary 4.6 and Theorem 4.5.

7.2 Equivariant families of convex subsets and their skinningmeasures

Let I be an index set endowed with a left action of Γ. A family D “ pDiqiPI of subsets of Xor of

p

GX indexed by I is Γ-equivariant if γDi “ Dγi for all γ P Γ and i P I. We will denote

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by „ “ „D the equivalence relation on I defined by i „ j if and only if Di “ Dj and thereexists γ P Γ such that j “ γi. This equivalence relation is Γ-equivariant: for all i, j P I andγ P Γ, we have γi „ γj if and only if i „ j. We say that D is locally finite if for every compactsubset K in X or in

p

GX, the quotient set ti P I : Di XK ‰ Hu„ is finite.

Examples. (1) Fixing a nonempty proper closed convex subset D of X, taking I “ Γ withthe left action by translations pγ, iq ÞÑ γi, and setting Di “ iD for every i P Γ gives aΓ-equivariant family D “ pDiqiPI . In this case, we have i „ j if and only if i´1j belongs tothe stabiliser ΓD of D in Γ, and I„ “ ΓΓD. Note that γD depends only on the class rγs of γin ΓΓD. We could also take I 1 “ ΓΓD with the left action by translations pγ, rγ1sq ÞÑ rγγ1s,and D 1 “ pγDqrγsPI 1 , so that for all i, j P I 1, we have i „D 1 j if and only if i “ j, and besides,D 1 is locally finite if and only if D is locally finite. The following choices of D yield equivariantfamilies with different characteristics:

(a) Let γ0 P Γ be a loxodromic element with translation axisD “ Axγ0 . The family pγDqγPΓis locally finite and Γ-equivariant. Indeed, by Lemma 2.1, only finitely many elementsof the family pγDqγPΓΓD meet any given bounded subset of X.

(b) Let ` P GX be a geodesic line whose image under the canonical map GX Ñ ΓzGX hasa dense orbit in ΓzGX under the geodesic flow, and let D “ `pRq be its image. Thenthe Γ-equivariant family pγDqγPΓ is not locally finite.

(c) More generally, let D be a convex subset such that ΓDzD is compact. Then the familypγDqγPΓ is a locally finite Γ-equivariant family.

(d) Let ξ P B8X be a bounded parabolic limit point of Γ, and let H be any horoball in Xcentred at ξ. Then the family pγH qγPΓ is a locally finite Γ-equivariant family.

(2) More generally, let pDαqαPA be a finite family of nonempty proper closed convex subsetsof X, and for every α P A, let Fα be a finite set. Define I “

Ť

αPA Γ ˆ tαu ˆ Fα with theaction of Γ by left translation on the first factor, and for every i “ pγ, α, xq P I, let Di “ γDα.Then I„ “

Ť

αPA ΓΓDα ˆ tαu ˆ Fα and the Γ-equivariant family D “ pDiqiPI is locallyfinite if and only if the family pγDαqγPΓ is locally finite for every α P A. The cardinalities ofFα for α P A contribute to the multiplicities (see Section 12.2).

Let D “ pDiqiPI be a locally finite Γ-equivariant family of nonempty proper closed convexsubsets of X. Let Ω “ pΩiqiPI be a Γ-equivariant family of subsets of

p

GX, where Ωi is ameasurable subset of B1

˘Di for all i P I (the sign ˘ being constant). Then

rσ˘Ω “ÿ

iPI„

rσ˘Di |Ωi ,

is a well-defined Γ-invariant locally finite measure onp

GX, whose support is contained inG˘, 0X. Hence, the measure rσ˘Ω induces a locally finite measure on Γz

p

GX, denoted by σ˘Ω ,see for example [PauPS, §2.6], in particular for warnings concerning the fact that Γ does notalways act freely on GX. When Ω “ B1

˘D “ pB1˘DiqiPI , the measure rσ˘Ω is denoted by

rσ˘D “ÿ

iPI„

rσ˘Di .

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The measures rσ`D and rσ´D are respectively called the outer and inner skinning measures of D

onp

GX, and their induced3 measures σ`D and σ´D on Γz

p

GX are the outer and inner skinningmeasures of D on Γz

p

GX.

Example. Consider the Γ-equivariant family D “ pγDqγPΓΓx with D “ txu a singleton in X.With π˘ “ pP˘D |B8Xq

´1 : B1˘D Ñ B8X the homeomorphism ρ ÞÑ ρ˘, we have pπ˘q˚rσ˘D “ µ˘x

by Remark (1) in Section 7, and

σ˘D “µ˘x

|Γx|. (7.14)

3See for instance [PauPS, §2.6] for details on the definition of the induced measure when Γ may havetorsion.

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Chapter 8

Explicit measure computations forsimplicial trees and graphs of groups

In this Chapter, we compute skinning measures and Bowen-Margulis measures for some highlysymmetric simplicial trees X endowed with a nonelementary discrete subgroup Γ of AutpXq.The potentials F are supposed to be 0 in this Chapter, and we assume that the Pattersondensities pµ`x qxPV X and pµ´x qxPV X of Γ are equal, denoted by pµxqxPV X. As the study ofgeometrically finite discrete subgroups of AutpXq mostly reduces to the study of particular(tree) lattices (see Remark 2.12), we will assume that Γ is a lattice in this Chapter.

The results of these computations will be useful when we state special cases of the equidis-tribution and counting results in regular and biregular trees and, in particular, in the arith-metic applications in Part III. The reader only interested in the continuous time case mayskip directly to Chapter 9.

A rooted simplicial tree pX, x0q is spherically symmetric if X is not reduced to x0 and hasno terminal vertex, and if the stabiliser of x0 in AutpXq acts transitively on each sphere ofcentre x0. The set of isomorphism classes of spherically symmetric rooted simplicial treespX, x0q is in bijection with the set of sequences ppnqnPN in N´t0u, where pn` 1 is the degreeof any vertex of X at distance n from x0.

x0

p0 ` 1 p2p1 p3 . . .

If pX, x0q is spherically symmetric, it is easy to check that the simplicial tree X is uniformif and only if the sequence ppnqnPN is periodic with palindromic period in the sense that thereexists N P N ´ t0u such that pn`N “ pn and pN´n “ pn for all n P N (such that n ď N for

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the second property). If N “ 1, then X “ Xp0 is the regular tree of degree p0 ` 1, and ifN “ 2, then X “ Xp0, p1 is the biregular tree of degrees p0 ` 1 and p1 ` 1.

The Hausdorff dimension hX of B8X for any visual distance is then

hX “1

Nlnpp0 . . . pN´1q ,

see for example [Lyo, p. 935].

8.1 Computations of Bowen-Margulis measures for simplicialtrees

The next result gives examples of computations of the total mass of Bowen-Margulis measuresfor lattices of simplicial trees having some regularity properties.

Analogous computations can be performed for Riemannian manifolds having appropriateregularity properties. We refer for instance to [PaP16a, Prop. 10] and [PaP16c, Prop. 20(1)] for computations of Bowen-Margulis measures for lattices in the isometry group of thereal hyperbolic spaces, and to [PaP16b, Lem. 4.2 (iii)] for the computation in the complexhyperbolic case. In both cases, the main point is the computation of the proportionalityconstant between the Bowen-Margulis measure and Sasaki’s Riemannian volume of the unittangent bundle. When dealing now with simplicial trees, similar consequences of homogeneityproperties will appear below.

We refer to Section 2.7 for the definitions of Tπ, Tvol, TVol appearing in the followingresult.

Proposition 8.1. Let pX, x0q be a spherically symmetric rooted simplicial tree with associatedsequence ppnqnPN such that X is uniform, and let Γ be a lattice of X.(1) For every x P V X, let rx “ dpx,AutpXqx0q, and let

cx “pprx ´ 1qe2 rx hX

pp0 ` 1q2p21 . . . p

2rx´1prx

`2p0

pp0 ` 1q2

if rx ‰ 0 and cx “ p0

p0`1 if rx “ 0. Then

mBM “ÿ

rxsPΓzV X

1

|Γx|

`

µx2 ´

ÿ

ePEX : opeq“x

µxpBeXq2˘

“ µx02

ÿ

rxsPΓzV X

cx|Γx|

. (8.1)

(2) If X “ Xp, q is the biregular tree of degrees p ` 1 and q ` 1, with V X “ VpX \ VqX thecorresponding partition of the set of vertices of X, if the Patterson density pµxqxPV X of Γ isnormalised so that µx “ p`1

?p for all x P VpX, then

pTπq˚mBM “ TvolΓzzX

andmBM “ TVolpΓzzXq “

ÿ

rxsPΓzVpX

p` 1

|Γx|`

ÿ

rxsPΓzVqX

q ` 1

|Γx|. (8.2)

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(3) If X “ Xq is the regular tree of degree q ` 1, if the Patterson density pµxqxPV X of Γ isnormalised to be a family of probability measures, then

π˚mBM “q

q ` 1volΓzzX

and in particularmBM “

q

q ` 1VolpΓzzXq . (8.3)

Proof. Let us first prove the first equality of Assertion (1). For every x P V X, we maypartition the set of geodesic lines ` P GX with `p0q “ x according to the two edges startingfrom x contained in the image of `. The only restriction for the edges is that they are requiredto be distinct.

For every e P EX, recall from Section 2.7 that BeX is the set of points at infinity of thegeodesic rays whose initial edge is e. For all e P EX and x P V X, say that e points away fromx if opeq P rx, tpeqs, and that e points towards x otherwise. In particular, all edges with originx point away from x. Hence by Equation (4.11), and since µx “ µ´x “ µ`x , we have

π˚mBM “ÿ

rxsPΓzV X

1

|Γx|

ÿ

e, e1PEX : opeq“ope1q“x, e‰e1

µ´x pBeXq µ`x pBe1Xq ∆rxs (8.4)

“ÿ

rxsPΓzV X

1

|Γx|

´

`

ÿ

ePEX : opeq“x

µxpBeXq˘2´

ÿ

ePEX : opeq“x

µxpBeXq2¯

∆rxs . (8.5)

This gives the first equality of Assertion (1).

Let us prove the second equality of Assertion (1). By homogeneity, we assume thatµx0 “ 1 and we will prove that

mBM “ÿ

rxsPΓzV X

cx|Γx|

.

Let N P N ´ t0u be such that pn`N “ pn and pN´n “ pn for all n P N, which exists sinceX is assumed to be uniform. Then the automorphism group AutpXq of the simplicial treeX acts transitively on the set of vertices at distance a multiple of N from x0. Hence forevery x P V X, the distance rx “ dpx,AutpXqx0q belongs to t0, 1, . . . , tN2 uu, and there existsγx, γ

1x P AutpXq such that dpx, γxx0q “ rx, x P rγxx0, γ

1xx0s and dpγxx0, γ

1xx0q “ N . The map

x ÞÑ rx is constant on the orbits of Γ in V X (actually on the orbits of AutpXq) and hence theright hand side of Equation (8.1) is well defined.

Since the family pµHausx qxPV X of Hausdorff measures of the visual distances pB8X, dxq is

invariant under any element of AutpXq, since Γ is a lattice and by Proposition 4.14, we haveδΓ “ hX and γ˚µx “ µγx for all x P V X and γ P AutpXq.

Since pX, x0q is spherically symmetric, and since µx0 is a probability measure, we have byinduction, for every e P EX pointing away from x0 with dpx0, opeqq “ n,

µx0pBeXq “1

pp0 ` 1q p1 . . . pn(8.6)

if n ‰ 0, and µx0pBeXq “ 1p0`1 otherwise.

For every fixed x P V X, let us now compute µxpBeXq for every edge e of X with origin x.Let γ “ γx, γ

1 “ γ1x P AutpXq be as above. By the spherical transitivity, we may assume thate or e belongs to the edge path from γx0 to γ1x0.

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rx

γx0

ex γ1x0

N ´ rx

There are two cases to consider.

Case 1: Assume first that e points away from γx0. There are p0 ` 1 such edges startingfrom x if rx “ 0, and prx otherwise. By Equation (8.6) and by invariance under AutpXq ofpµHausx qxPV X, we have

µγx0pBeXq “1

pp0 ` 1q p1 . . . prx,

with the convention that the denominator is p0 ` 1 if rx “ 0. Since the map ξ ÞÑ βξpx, γx0q

is constant with value ´rx on BeX, and by the quasi-invariance property of the Pattersondensity (see Equation (4.2)), we have

µxpBeXq “ e´δΓp´rxqµγx0pBeXq “erxhX

pp0 ` 1q p1 . . . prx,

with the same convention as above.

Case 2: Assume now that e points towards γx0. This implies that rx ě 1, and there is oneand only one such edge starting from x. Then as above we have

µγ1x0pBeXq “1

ppN ` 1q pN´1 . . . prx,

and

µxpBeXq “ e´δΓp´pN´rxqqµγ1x0pBeXq “epN´rxqhX

ppN ` 1q pN´1 . . . prx.

Therefore, if we set for every x P V X,

Cx “´

ÿ

ePEX : opeq“x

µxpBeXq¯2´

ÿ

ePEX : opeq“x

µxpBeXq2 , (8.7)

we have if rx ‰ 0, since eNhX “ p0p1 . . . pN´1 and pN “ p0,

Cx “´

prxerxhX

pp0 ` 1q p1 . . . prx`

epN´rxqhX

ppN ` 1q pN´1 . . . prx

¯2

´

´

prx` erxhX

pp0 ` 1q p1 . . . prx

˘2`` epN´rxqhX

ppN ` 1q pN´1 . . . prx

˘2¯

“pprx

2´ prxq e

2 rxhX

pp0 ` 1q2 p12 . . . prx

2`

2 prxeN hX

pp0 ` 1q p1 . . . prxprx . . . pN´1ppN ` 1q

“pprx ´ 1q e2 rxhX

pp0 ` 1q2 p12 . . . prx´1

2prx`

2 p0

pp0 ` 1q2“ cx ,

and, if rx “ 0,

Cx “´

pp0 ` 1q1

p0 ` 1

¯2´ pp0 ` 1q

´ 1

p0 ` 1

¯2“

p0

p0 ` 1“ cx .

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Assertion (1) of Proposition 8.1 now follows from Equation (8.5).

Let us prove Assertion (2) of Proposition 8.1. Note that X “ Xp, q is spherically symmetricwith respect to any vertex of X, and that

hX “1

2lnppqq .

Let e be an edge of X, with x “ opeq P VpX and y “ tpeq P VqX. For every z P V X, we defineCz as in Equation (8.7).

Note that by homogeneity, we have Cz “ Cx and µz “ µx for all z P VpX, as well asCz “ Cy and µz “ µy for all z P VqX. Hence the normalisation of the Patterson densityas in the statement of Assertion (2) is possible. By the spherical symmetry at x, and thenormalisation of the measure, we have µxpBeXq “ 1?

p and µxpBeXq “?p. Therefore

µy “ µypBeXq ` µypBeXq “ ehXµxpBeXq ` e´hXµxpBeXq

“?pq

1?p`

1?pq

?p “

q ` 1?q

.

This symmetry in the values of µy and µx explains the choice of our normalisation. Wehave

Cx “ µx2 ´ pp` 1q

´

µx

p` 1

¯2“

p

p` 1µx

2 “ p` 1

and similarly Cy “ qq`1µy

2 “ q ` 1. This proves the second equality in Equation (8.2), bythe first equation of Assertion (1).

In order to prove that pTπq˚mBM “ TvolΓzzX, we now partition ΓzGX asď

resPΓzEXt` P ΓzGX : `p0q “ πpopeqq, `p1q “ πptpeqqu .

Using on every element of this partition Hopf’s decomposition with respect to the basepointopeq, we have, by a ramified covering argument already used in the proof of the second partof Proposition 4.13,

pTπq˚mBM “ÿ

resPΓzEX

1

|Γe|µopeqpB8X´ BeXq µopeqpBeXq ∆res .

Since µopeqpBeXq “ e´hXµtpeqpBeXq and by homogeneity, we have

pTπq˚mBM “ÿ

resPΓzEX

1

|Γe|

deg opeq ´ 1

deg opeqµopeq

deg tpeq ´ 1

deg tpeqµtpeq e

´hX ∆res

“ÿ

resPΓzEX

1

|Γe|

µopeq µtpeq?pq

pp` 1qpq ` 1q∆res “ TvolΓzzX .

The first equality of Equation (8.2) follows.

Finally, the last claim of Assertion (3) of Proposition 8.1 follows from Equation (8.1), sincecx “

qq`1 for every x P V Xq (or by taking q “ p in Equation (8.2) and by renormalising). The

first claim of Assertion (3) follows from the first claim of Assertion (2), by using Equation(2.16). l

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Remark 8.2. (1) In particular, when X “ Xq is regular, the Patterson density is normalisedto be a family of probability measures and Γ is torsion free, then π˚mBM is q

q`1 times thecounting measure on ΓzV X. In this case, Equation (8.3) is given by [CoP4, Rem. 2].(2) If X “ Xp, q is biregular with p ‰ q, then π˚mBM is not proportional to volΓzzX. Inparticular, if Γ is torsion free and the Patterson density is normalised to be a family ofprobability measures, then π˚mBM is the sum of p

p`1 times the counting measure on ΓzVpXand q

q`1 times the counting measure on ΓzVqX.This statement is similar to the well-known fact that in pinched but variable curvature,

the Bowen-Margulis measure is generally not absolutely continuous with respect to Sasaki’sRiemannian measure on the unit tangent bundle (it would then be proportional by ergodicityof the geodesic flow in the lattice case).

8.2 Computations of skinning measures for simplicial trees

We now give examples of computations of the total mass of skinning measures (for zeropotentials), after introducing some notation. Let X be a locally finite simplicial tree withoutterminal vertices, and let Γ be a discrete subgroup of AutpXq.

For every simplicial subtree D of X, we define the boundary BV D of V D in X as

BV D “ tx P V D : D e P EX, opeq “ x, tpeq R V Du .

The boundary BD of D is the maximal subgraph (which might not be connected) of X withset of vertices BV D. If Γ is a discrete subgroup of AutpXq, then the stabiliser ΓD of D actsdiscretely on BD.

For every x P V X, we define the codegree of x in D as codegDpxq “ 0 if x R D and otherwise

codegDpxq “ degXpxq ´ degDpxq .

Note that codegDpxq “ 0 if x R BV D, and that the codegree codegN1Dpxq of x P V X is 0unless x lies in the boundary of the 1-neighbourhood of D, in which case it is constant equalto degXpxq ´ 1.

Let D “ pDiqiPI be a locally finite Γ-equivariant family of simplicial subtrees of X, and letx P V X. We define the multiplicity1 of x in (the boundary of) D as (see Section 7.2 for thedefinition of „D)

mDpxq “Cardti P I„D : x P BV Diu

|Γx|.

The numerator and the denominator are finite, by the local finiteness of the family D andthe discreteness of Γ, and they depend only on the orbit of x under Γ. Note that if D is asimplicial subtree of X which is precisely invariant under Γ (that is, whenever γ P Γ is suchthat DX γD is nonempty, then γ belongs to the stabiliser ΓD of D in Γ), if D “ pγDqγPΓΓD ,and if x P BV D, then

mDpxq “1

|Γx|.

In particular, if furthermore Γ is torsion free, then mDpxq “ 1 if x P BV D, and mDpxq “ 0otherwise.

1See Section 12.2 for explanations on the terminology.

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Example 8.3. Let G be a connected graph without vertices of degree ď 2 and let X be itsuniversal cover, with covering group Γ. If C is a cycle in G “ ΓzX and if D is the family ofgeodesic lines in X lifting C, then mDpxq “ 1 for all x P V X whose image in G “ ΓzX belongsto C if C is a simple cycle (that is, if C passes through no vertex twice).

We define the codegree of x in D as

codegDpxq “ÿ

iPI„D

codegDipxq ,

which is well defined as codegDipxq depends only on the class of i P I modulo „D . Note that

codegDpxq “ pdegX x´ kq |Γx|mDpxq (8.8)

if degDipxq “ k for every x P BV Di and i P I. If every vertex of X has degree at least 3, this isin particular the case with k “ 2 if Di is a line for all i P I and with k “ 1 if Di is a horoballfor all i P I.

We will say that a simplicial subtree D of X, with stabiliser ΓD in Γ, is almost preciselyinvariant if there exists N P N such that for every x P BV D, the number of γ P ΓΓD suchthat x P γBV D is at most N . It follows from this property that if D “ pγDqγPΓ, then D islocally finite and codegDpxq ď N codegDpxq for every x P X.

Proposition 8.4. Assume that X is a regular or biregular simplicial tree with degrees at least3, and that Γ is a lattice of X.(1) For every simplicial subtree D of X, we have

π˚rσ˘D “

ÿ

xPV X

µx codegDpxq

degXpxq∆x .

(2) If D “ pDiqiPI is a locally finite Γ-equivariant family of simplicial subtrees of X, then

π˚σ˘D “

ÿ

rxsPΓzV X

µx codegDpxq

|Γx| degXpxq∆rxs .

(3) Let k P N and let D be a simplicial subtree of X such that degDpxq “ k for every x P BV Dand the Γ-equivariant family D “ pγDqΓΓD is locally finite. Then

π˚σ˘D “

ÿ

ΓDyPΓDzBV D

µy pdegXpyq ´ kq

|pΓDqy| degXpyq∆Γy .

(4) If D is a simplicial subtree of X such that the Γ-equivariant family D “ pγDqΓΓD is locallyfinite, then the skinning measure σ˘D is finite if and only if the graph of groups ΓDzzBD hasfinite volume.

Before proving Proposition 8.4, let us give some immediate consequences of Assertion (3).If X “ Xp, q is biregular of degrees p` 1 and q` 1, let V X “ VpX \ VqX be the correspondingpartition of the set of vertices of X and, for r P tp, qu, let BrD be the edgeless graph with setof vertices BV DX VrX.

117 19/12/2016

Corollary 8.5. Assume that pX,Γq is as in Proposition 8.4. Let D be a simplicial subtree ofX such that the Γ-equivariant family D “ pγDqΓΓD is locally finite.(1) If X “ Xp, q is biregular of degrees p` 1 and q ` 1 and if the Patterson density pµxqxPV X

of Γ is normalised so that µx “degXpxq?degXpxq´1

for all x P V X, then

‚ if D is a horoball,

σ˘D “?p VolpΓDzzBpDq `

?q VolpΓDzzBqDq ,

‚ if D is a line,

σ˘D “p´ 1?p

VolpΓDzzBpDq `q ´ 1?q

VolpΓDzzBqDq . (8.9)

(2) If X “ Xq is the regular tree of degree q ` 1 and if the Patterson measures pµxqxPV X arenormalised to be probability measures, then‚ if D is a horoball,

σ˘D “q

q ` 1VolpΓDzzBDq (8.10)

‚ if D is a line,

σ˘D “q ´ 1

q ` 1VolpΓDzzDq . (8.11)

Proof of Proposition 8.4. (1) We may partition the outer/inner unit normal bundleB1˘D of D according to the first/last edge of the elements in B1

˘D. On each of the elementsof this partition, for the computation of the skinning measures using its definition and itsindependence of the basepoint (see Section 7.1), we take as basepoint the initial/terminalpoint of the corresponding edge. Since D is a simplicial tree, note that for every e P EX suchthat opeq P V D, we have e P ED if and only if tpeq P V D. Thus, we have

π˚rσ`D “

ÿ

ePEX : opeqPV D, tpeqRV DµopeqpBeXq ∆opeq

“ÿ

xPBV D

´

ÿ

ePEX : opeq“x, tpeqRV DµxpBeXq

¯

∆x .

and similarly

π˚rσ´D “

ÿ

ePEX : tpeqPV D, opeqRV DµtpeqpBeXq ∆tpeq

“ÿ

xPBV D

´

ÿ

ePEX : opeq“x, tpeqRV DµxpBeXq

¯

∆x .

As in the proof of Proposition 8.1 (2), since X is spherically homogeneous around each pointand since Γ is a lattice (so that the Patterson density is AutpXq-equivariant, see Proposition4.14), we have µxpBeXq “ µx

degXpxqfor all x P V X and e P EX with opeq “ x. Assertion

(1) of Proposition 8.4 follows, sinceř

ePEX : opeq“x, tpeqRV D 1 “ codegDpxq if x P BV D andcodegDpxq “ 0 otherwise.

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(2) By the definition2 of the skinning measures associated with Γ-equivariant families, wehave rσ˘D “

ř

iPI„rσ`Di , where „ “ „D . Hence by Assertion (1)

π˚rσ˘D “

ÿ

iPI„

rπ˚σ`Di “

ÿ

iPI„

ÿ

xPV X

µx codegDipxq

degXpxq∆x

“ÿ

xPV X

´

ÿ

iPI„

codegDipxq¯

µx

degXpxq∆x “

ÿ

xPV X

µx codegDpxq

degXpxq∆x .

By the definition of the measure induced in ΓzV X when Γ may have torsion (see for instance[PauPS, §2.6]), Assertion (2) follows.

(3) It follows from Assertion (2) and from Equation (8.8) that

π˚σ˘D “

ÿ

rxsPΓzV X

degXpxq ´ k

degXpxqµx mDpxq ∆rxs .

For every x P V X, by the definition of mDpxq, we have, by partitioning BV D into its orbitsunder ΓD,

mDpxq “1

|Γx|Cardtγ P ΓDzΓ : γx P BV Du

“1

|Γx|

ÿ

ΓDyPΓDzBV DCardtγ P ΓDzΓ : ΓDγx “ ΓDyu

“1

|Γx|

ÿ

ΓDyPΓDzBV D, Γx“Γy

Cardtγ P ΓDzΓ : ΓDγy “ ΓDyu

“1

|Γx|

ÿ

ΓDyPΓDzBV D, Γx“Γy

rΓy : pΓDqys “ÿ

ΓDyPΓDzBV D, Γx“Γy

1

|pΓDqy|.

This proves Assertion (3), sinceř

rxsPΓzV X, Γx“Γy ∆rxs “ ∆Γy.

(4) It follows from Assertion (2) that

σ˘D “ÿ

rxsPΓzV X

µx codegDpxq

|Γx| degXpxq.

Note that for every x P BV D, we have

|Γx|mDpxq ď codegDpxq ď degXpxq |Γx|mDpxq .

Let m “ minxPV X µx and M “ maxxPV X µx, which are positive and finite, as the totalmass of the Patterson measures takes at most two values, since Γ is a lattice and X is biregular.By arguments similar to those in the proof of Assertion (3), we hence have

m

minxPV X degXpxqVolpΓDzzBDq ď σ˘D ďM VolpΓDzzBDq

The result follows. l

2See Section 7.2.

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We now give a formula for the skinning measure (with zero potential) of a geodesic linein the simplicial tree X, using Hamenstädt’s distance dH and measure3 µH associated witha fixed horoball H in X. This expression for the skinning measure will be useful in Part III.

Lemma 8.6. Let H be a horoball in X centred at a point ξ P B8X. Let L be a geodesic linein X with endpoints L˘ P B8X´ tξu. Then for all ρ P B1

`L such that ρ` ‰ ξ,

drσ`L pρq “dH pL`, L´q

δΓ

dH pρ`, L´qδΓ dH pρ`, L`qδΓdµH pρ`q .

Proof. By Equations (2.9) and (7.6), the power dHδΓ of the distance and the measure µH

scale by the same factor when the horoball is replaced by another one centred at the samepoint. Thus, we can assume in the proof that L does not intersect the interior of H .

Fix ρ P B1`L such that ρ` ‰ ξ. Let y be the closest point to ξ on L, let x0 be the closest

point to L on H , and let z be the closest point to ξ on ρpr0,`8rq. Let t ÞÑ xt be the geodesicray starting from x0 at time t “ 0 and converging to ξ. When t is big enough, the points ρ`,z, xt and ξ are in this order on the geodesic line sρ`, ξr.

We have, by the definition in Equation (7.1) of the skinning measure,

drσ`L pρq “ eδΓ βρ` pxt, ρp0qqdµxtpρ`q “ eδΓ βρ` pxt, zq`δΓ βρ` pz, ρp0qqdµxtpρ`q

“ e´δΓ βξpxt, zq´δΓ dpz, ρp0qqdµxtpρ`q “ eδΓ t´δΓ βξpx0, zq´δΓ dpz, ρp0qqdµxtpρ`q ,

and by the definition of Hamenstädt’s measure µH (see Equation (7.5))

dµH pρ`q “ eδΓ tdµxtpρ`q .

ρ`

yL´

x0

L`

xt

z “ ρp0q

BH

ξ

L´ L`

xtBH

ξ

ρ`

y “ ρp0q

x0 “ z

Case 1: Assume first that ρp0q ‰ y. We may assume that ρp0q P ry, L`r. Then z “ ρp0q andz is the closest point to H on the geodesic line sL`, ρ`r. Thus dH pL´, L`q “ dH pL´, ρ`qand dH pL`, ρ`q “ e´dpz, x0q “ eβξpx0, zq, and the claim follows.

Case 2: Assume now that y “ ρp0q. Then ry, zs “ ry, ξr X ry, ρ`r, and we may assume thatx0 “ z up to adjusting the horoball H while keeping its point at infinity. Thus dH pL´, L`q “e´dpy, x0q “ e´dpz, ρp0qq and dH pL´, ρ`q “ dH pL`, ρ`q “ 1, and the claim follows. l

3See the definitions of Hamenstädt’s distance and measure in Sections 2.3 and 7.1 respectively.

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Chapter 9

Rate of mixing for the geodesic flow

Let X,x0,Γ, rF , pµ˘x qxPX be as in the beginning of Chapter 7. In this Chapter, we start

by collecting in Section 9.1 known results on the rate of mixing of the geodesic flow formanifolds. The main part of the Chapter then consists in proving analogous bounds for thediscrete time and continuous time geodesic flow for quotient spaces of simplicial and metrictrees respectively.

We define mF “mFmF

when the Gibbs measure is finite. Recall that this measure isnonzero since Γ is nonelementary.

Let α P s0, 1s.1 We will say that the (continuous time) geodesic flow on ΓzGX is ex-ponentially mixing for the α-Hölder regularity or that it has exponential decay of α-Höldercorrelations for the potential F if there exist C, κ ą 0 such that for all φ, ψ P C α

b pΓzGXq andt P R, we have

ˇ

ˇ

ˇ

ż

ΓzGXφ ˝ g´t ψ dmF ´

ż

ΓzGXφ dmF

ż

ΓzGXψ dmF

ˇ

ˇ

ˇď C e´κ|t| φα ψα ,

and that it is polynomially mixing or has polynomial decay of α-Hölder correlations if thereexist C ą 0 and n P N´ t0u such that for all φ, ψ P C α

b pΓzGXq and t P R, we have

ˇ

ˇ

ˇ

ż

ΓzGXφ ˝ g´t ψ dmF ´

ż

ΓzGXφ dmF

ż

ΓzGXψ dmF

ˇ

ˇ

ˇď C p1` |t|q´n φα ψα .

9.1 Rate of mixing for Riemannian manifolds

When X “ ĂM is a complete Riemannian manifold with pinched negative sectional curvaturewith bounded derivatives, then the boundary at infinity of ĂM , the strong unstable, unstable,stable, and strong stable foliations of T 1

ĂM are only Hölder-smooth in general.2 Hence Hölderregularity on functions on T 1

ĂM is appropriate.The geodesic flow is known to have exponential decay of Hölder correlations for compact

manifolds M when• M is two-dimensional and F is any Hölder potential by [Dol1],

1We refer to Section 3.1 for the definition of the Banach space C αb pZq of bounded α-Hölder-continuous

functions on a metric space Z.2See for instance [Brin] when ĂM has a compact quotient (a result first proved by Anosov), and [PauPS,

Theo. 7.3].

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• M is 19-pinched and F “ 0 by [GLP, Coro. 2.7],• mF is the Liouville measure by [Live], see also [Tsu], [NZ, Coro. 5] who give more precise

estimates,• M is locally symmetric and F is any Hölder potential by [Sto], see also [MO].

When ĂM is a symmetric space, then the boundary at infinity of ĂM , the strong unstable,unstable, stable, and strong stable foliations of T 1

ĂM are smooth. Hence talking about leafwiseC `-smooth functions on T 1

ĂM makes sense. We will denote by C `c pNq the vector space of real-

valued C `-smooth functions on the orbifold T 1M “ ΓzT 1ĂM (that is, the maps induced on

T 1M by the C `-smooth Γ-invariant functions on T 1ĂM), with compact support in T 1M , and

by ||ψ||` the Sobolev W `,2-norm of any ψ P C `c pT

1Mq.Given ` P N, we will say that the geodesic flow on T 1M is exponentially mixing for the `-

Sobolev regularity (or that it has exponential decay of `-Sobolev correlations) for the potentialF if there exist c, κ ą 0 such that for all φ, ψ P C `

c pT1Mq and all t P R, we have

ˇ

ˇ

ˇ

ż

T 1Mφ ˝ g´t ψ dmF ´

ż

T 1Mφ dmF

ż

T 1Mψ dmF

ˇ

ˇ

ˇď c e´κ|t| ψ` φ` .

When F “ 0 and Γ is an arithmetic lattice in the isometry group of ĂM (the Gibbs measurethen coincides, up to a multiplicative constant, with the Liouville measure), this property, forsome ` P N, follows from [KM1, Theorem 2.4.5], with the help of [Clo, Theorem 3.1] to checkits spectral gap property, and of [KM2, Lemma 3.1] to deal with finite cover problems.

9.2 Rate of mixing for simplicial trees

Let X be a locally finite simplicial tree without terminal vertices, with geometric realisationX “ |X|1. Let Γ be a nonelementary discrete subgroup of AutpXq and let rc : EX Ñ R be asystem of conductances for Γ on X.

In this Section, building on the end of Section 4.4 concerning the mixing properties them-selves, we now study the rates of the mixing properties of the discrete time geodesic flow onΓzGX for the Gibbs measure mc “ mFc , when it is mixing.

Let pZ,m, T q be a dynamical system with pZ,mq a probability space and T : Z Ñ Z a(not necessarily invertible) measure preserving map. For all n P N and φ, ψ P L2pmq, the(well-defined) n-th correlation coefficient of φ, ψ is

covm,npφ, ψq “

ż

Zφ ˝ Tn ψ dm´

ż

Zφ dm

ż

Zψ dm .

Let α P s0, 1s and assume that Z is a metric space (endowed with its Borel σ-algebra).Similarly as for the case of flows in the beginning of Chapter 9, we will say that the dynamicalsystem pZ,m, T q is exponentially mixing for the α-Hölder regularity or that it has exponentialdecay of α-Hölder correlations if there exist C, κ ą 0 such that for all φ, ψ P C α

b pZq and n P N,we have

| covm,npφ, ψq| ď C e´κn φα ψα .

Note that this property is invariant under measure preserving conjugations of dynamicalsystems by bilipschitz homeomorphisms.

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The main result of this Section is a simple criterion for the exponential decay of correlationfor the discrete time geodesic flow on ΓzGX.

We define mc “mcmc

when the Gibbs measure mc on ΓzGX is finite, and we use thedynamical system pΓzGX, mc, g

1q in the definition of the correlation coefficients.Given a finite subset E of ΓzV X, we denote by τE : ΓzGX Ñ N Y t`8u the first return

time to E of the discrete time geodesic flow:

τEp`q “ inftn P N´ t0u : gn`p0q P Eu ,

with the usual convention that infH “ `8.

Theorem 9.1. Let X,Γ,rc be as above, with δc finite. Assume that the Gibbs measure mc isfinite and mixing for the discrete time geodesic flow on ΓzGX. Assume moreover that thereexist a finite subset E of ΓzV X and C 1, κ1 ą 0 such that for all n P N, we have

mc

`

t` P ΓzGX : `p0q P E and τEp`q ě nu˘

ď C 1 e´κ1n . (9.1)

Then the discrete time geodesic flow on ΓzGX has exponential decay of α-Hölder correlationsfor the system of conductances c.

A similar statement holds for the square of the discrete time geodesic flow on ΓzGevenXwhen mc is finite, C ΛΓ is a uniform simplicial tree with degrees at least 3 and LΓ “ 2Z.

Note that the crucial Hypothesis (9.1) of Theorem 9.1 is in particular satisfied if ΓzXis finite, by taking E “ ΓzV X. But the result is quite well-known in this case: when Γ istorsion free, it follows from Bowen’s result [Bowe2, 1.26] that a mixing subshift of finite typeis exponentially mixing.

Proof. Let X1 “ C ΛΓ. Using the coding introduced in Section 5.2, we first reduce thisstatement to a symbolic dynamics one.

Step 1 : Reduction to two-sided symbolic dynamics

Let pΣ, σq be the (two-sided) topological Markov shift with alphabet A and transitionmatrix A constructed in Section 5.2, conjugated to pΓzGX1, g1q by the homeomorphism Θ :ΓzGX1 Ñ Σ (see Theorem 5.1). Let P “ Θ˚

mcmc

, which is a mixing σ-invariant probabilitymeasure on Σ. Let

E “ tpe´, h, e`q P A : tpe´q “ ope`q P Eu .

The set E is finite since the degrees and the vertex stabilisers of X are finite. For all x P Σand k P Z, we denote by xk the k-th component of x “ pxnqnPZ. Let

τE pxq “ inftn P N´ t0u : xn P E u

be the first return time to E of x under iteration of the shift σ.Let π` : Σ Ñ A N be the natural extension pxnqnPZ ÞÑ pxnqnPN. Theorem 9.1 will follow

from the following two-sided symbolic dynamics result.3

Theorem 9.2. Let pΣ, σq be a locally compact transitive two-sided topological Markov shiftwith alphabet A and transition matrix A, and let P be a mixing σ-invariant probability measurewith full support on Σ. Assume that

3Assumption (1) of Theorem 9.2 is far from being optimal, but will be sufficient for our purpose.

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(1) for every A-admissible finite sequence pw0, . . . , wnq in A , the Jacobian of the map fromtpxkqkPN P π`pΣq : x0 “ wnu to tpykqkPN P π`pΣq : y0 “ w0, . . . , yn “ wnudefined by px0, x1, x2, . . . q ÞÑ pw0, . . . , wn, x1, x2, . . . q, with respect to the restrictions ofthe pushforward measure pπ`q˚P, is constant;

(2) there exist a finite subset E of A and C 1, κ1 ą 0 such that for all n P N, we have

P`

tx P Σ : x0 P E and τE pxq ě nu˘

ď C 1 e´κ1n . (9.2)

Then pΣ,P, σq has exponential decay of α-Hölder correlations.

Proof that Theorem 9.2 implies Theorem 9.1. Since rmc is supported on GX1, up toreplacing X by X1, we may assume that B8Γ “ ΛΓ.

By the construction of Θ just before the statement of Theorem 5.1, for every ` “ Γr` PΓzGX, we have pΘ`q0 “ pe´0 pr`q, h0pr`q, e

`0 p

r`qq with e`0 pr`qq “ ppr`pr0, 1sqq where p : XÑ ΓzX1 isthe canonical projection, so that o

`

e`0 pr`qq

˘

“ `p0q. Since Θ conjugates g1 to σ, we have

pΘ`qn “ pσnpΘ`qq0 “ pΘpg

n`qq0 P E

if and only if gn`p0q P E, andτE pΘ`q “ τEp`q .

Therefore Theorem 9.1 will follow from Theorem 9.2 by conjugation since Θ is bilipschitz,once we have proved that Hypothesis (1) of Theorem 9.2 is satisfied for the two-sided topo-logical Markov shift pΣ, σq conjugated by Θ to pΓzGX, g1q, which is the main point in thisproof.

We hence fix an A-admissible finite sequence w “ pw0, . . . , wnq in A . We denote by

rwns “ tpxkqkPN P π`pΣq : x0 “ wnu ,

rws “ tpykqkPN P π`pΣq : y0 “ w0, . . . , yn “ wnu

and fw : px0, x1, x2, . . . q ÞÑ pw0, . . . , wn, x1, x2, . . . q the sets and map appearing in Hypothesis(1). We denote by rw and Ăwn the discrete generalised geodesic lines in X associated with wand wn (see the proof of Theorem 5.1 just after Equation (5.3)). Since w ends with wn, bythe construction of Θ, there exists γ P Γ sending the two consecutive edges of rwn to the lasttwo consecutive edges of w. We denote by x “ rwp0q and y “ Ăwnp0q the footpoints of rw andĂwn respectively.

rw

Ăwn

γyx “ rwp0q

y “ Ăwnp0q

η

ξ

γ´1ξ

γ´1η

γ

124 19/12/2016

For every discrete generalised geodesic line ω P

p

GX which is isometric exactly on aninterval I containing 0 in its interior (as for ω “ rw,Ăwn), let

GωX “ t` P GX : `|I “ ω|Iu

be the space of extensions of ω|I to geodesic lines. With B˘ωX “ t`˘ : ` P GωXu its setof points at ˘8, we have a homeomorphism GωX Ñ pB´ωX ˆ B`ωXq defined by ` ÞÑ p`´, ``q,using Hopf’s parametrisation with respect to the point ωp0q, since all the geodesic lines inGωX are at the point ωp0q at time t “ 0. Using as basepoint x0 “ ωp0q in the definition of theGibbs measure (see Equation (4.10)), this homeomorphism sends the restriction to GωX of theGibbs measure d rmcp`q to the product measure dµ´ωp0qp`´q dµ

`

ωp0qp``q. Hence the pushforwardof rmc|GωX by the positive endpoint map e` : ` ÞÑ `` is µ´ωp0qpB

´ωXq dµ`ωp0qp``q, and note that

µ´ωp0qpB´ωXq is a positive constant.

Since π` : Σ Ñ Σ` is the map which forgets about the past, there exist measurable mapsuw : B`

rwXÑ rws and uwn : B`ĂwnXÑ rwns such that the following diagrams commute:

GrwX

π` ˝Θ ˝ p

ÝÝÝÝÑ rws

e`Œ Õuw

B`rwX

and

GĂwnX

π` ˝Θ ˝ p

ÝÝÝÝÑ rwns

e`Œ Õuwn

B`

ĂwnX

.

Furthermore, the map uw (respectively uwn) is surjective, and has constant finite order fibersgiven by the orbits of the finite stabiliser Γ

rw (respectively ΓĂwn). Since P “ Θ˚

mcmc

, the push-forward by the map uw (respectively uwn) of the measure µ`x (respectively µ`y ) is a constanttime the restriction of pπ`q˚P to rws (respectively rwns). Finally, by the construction of the(inverse of the) coding in the proof of Theorem 5.1, the following diagram is commutative:

B`

ĂwnX uwn

ÝÑ rwns

γ Ó Ófw

B`rwX

uwÝÑ rwns .

Recall that the pushforwards of measures µ, ν, which are absolutely continuous one withrespect to the other, by a measurable map f are again absolutely continuous one with respectto the other, and satisfy (almost everywhere)

d f˚µ

d f˚ν˝ f “

dµ

d ν.

Hence in order to prove that Hypothesis (1) in the statement of Theorem 9.2 is satisfied, weonly have to prove that the map γ : B`

ĂwnX Ñ B

`rwX has a constant Jacobian for the measures

µ`y on B`ĂwnX and µ`x on B`

rwX respectively.For all ξ, η P B`

rwX, by the properties of the Patterson densities (see Equations (4.1) and(4.2)), and since γy belongs to the geodesic ray from x to ξ and η (see Equation (3.6) andthe above picture), we have

d γ˚µ`y

dµ`xpξq

d γ˚µ`y

dµ`xpηq

“

dµ`γydµ`x

pξq

dµ`γydµ`x

pηq“e´C

`ξ pγy, xq

e´C`η pγy, xq

“e´

şγyx p

rF`c ´δcq

e´şγyx p

rF`c ´δcq“ 1 .

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This proves that Hypothesis (1) in Theorem 9.2 is satisfied, and concludes the proof ofTheorem 9.1. l

We now indicate how to pass from a one-sided version of Theorem 9.2 to the two-sidedone, as was communicated to us by J. Buzzi.

Step 2 : Reduction to one-sided symbolic dynamics

Let pΣ`, σ`q be the one-sided topological Markov shift with alphabet A and transitionmatrix A, that is, Σ` is the closed subset of the topological product space A N defined by

Σ` “

x “ pxnqnPN P A N : @ n P N, Axn,xn`1 “ 1u ,

and σ` : Σ` Ñ Σ` is the (one-sided) shift4 defined by

pσ`pxqqn “ xn`1

for all x P Σ` and n P N. We endow Σ` with the distance

dpx, x1q “ e´max

nPN : @ i P t0,...,nu, xi “ x1i

(

.

Note that the distances on Σ and Σ` are bounded by 1.Let π` : Σ Ñ Σ` be the natural extension pxnqnPZ ÞÑ pxnqnPN, which satisfies π` ˝ σ “

σ` ˝π` and is 1-Lipschitz. Note that Σ is transitive (respectively locally compact) if and onlyif Σ` is transitive (respectively locally compact).

In the one-sided case, we always assume that the cylinders start at time t “ 0: given anadmissible sequence w “ pw0, w1, . . . , wn´1q, the cylinder of length |w| “ n it defines is

rws “ rw0, . . . , wn´1s “ tpxnqnPN P Σ` : @ i P t0, . . . , n´ 1u, xi “ wiu .

We first explain how to relate the decay of correlations for the two-sided and one-sidedsystems. This is well-known since the works of Sinai [Sin, §3] and Bowen [Bowe2, Lem. 1.6], seefor instance [You1, §4], and the following proof has been communicated to us by J. Buzzi. Wefix α P s0, 1s. For all metric space Z and bounded α-Hölder-continuous function f : Z Ñ R,let

f1α “ supx, yPZ

0ădpx,yqď1

|fpxq ´ fpyq|

dpx, yqα,

so that5 fα “ f8 ` f1α.

Lemma 9.3. For every a P A , let us fix za P Σ such that pzaq0 “ a. Let φ : Σ Ñ R be abounded α-Hölder-continuous map and N P N. Define φpNq : Σ` Ñ R by:

@x P Σ`, φpNqpxq “ φpyq where"

yi “ xi`N if i ě ´Nyi “ zx0

i`N otherwise.

Then φpNq is bounded and α-Hölder-continuous on Σ`, with

|φ ˝ σN ´ φpNq ˝ π`| ď φ1α e

´αN .

Moreover,φpNq1α ď eαN φ1α and φpNq8 ď φ8 .

4Although it is standard to denote the one-sided shift by σ in the same way as the two-sided shift, we useσ` for readability.

5See Section 3.1 for the definition of the Hölder norm ¨ α.

126 19/12/2016

Proof. For all x “ pxnqnPZ P Σ, with y associated with π`pxq as in the statement, we havepσN pxq

˘

n“ yn if |n| ď N , hence

|φ ˝ σN pxq ´ φpNqpπ`pxqq| “ |φpσN pxqq ´ φpyq| ď φ1α dpσ

N pxq, yqα ď φ1α e´αN .

Moreover, if y, y1 P Σ are associated with x “ pxnqnPN, x1 “ px1nqnPN P Σ` respectively, thendpy, y1q “ eN dpx, x1q if dpx, x1q ă e´N and otherwise dpy, y1q ď 1 ď eNdpx, x1q, so that

|φpNqpxq ´ φpNqpx1q| “ |φpyq ´ φpy1q| ď φ1α dpy, y1qα ď φ1α e

αN dpx, x1qα . l

Proposition 9.4. Let µ be a σ-invariant probability measure on Σ. Assume that the dynam-ical system pΣ`, σ`, pπ`q˚µq has exponential decay of α-Hölder correlations. Then pΣ, σ, µqhas exponential decay of α-Hölder correlations.

Proof. Let C, κ ą 0 be such that for all bounded α-Hölder-continuous maps φ1, ψ1 : Σ` Ñ Rand n P N, we have

| covpπ`q˚µ, npφ1, ψ1q| ď C φα ψα e

´κn .

Let φ, ψ : Σ Ñ R be bounded α-Hölder-continuous maps and n P N. Denoting by ˘ t anyvalue in r´t, ts for any t ě 0, we have, by the first part of the above lemma and for any N P N(to be chosen appropriately later on),

ż

Σφ ˝ σn ψ dµ “

ż

Σφ ˝ σn`N ψ ˝ σN dµ

“

ż

ΣpφpNq ˝ π` ˘ φ

1α e

´αN q ˝ σn pψpNq ˝ π` ˘ ψ1α e

´αN q dµ

“

ż

Σ`

φpNq ˝ σn` ψpNq dpπ`q˚µ ˘ φα ψα e´αN .

A similar estimate holds for the second term in the definition of the correlation coefficients.Hence by the second part of the above lemma

| covµ, npφ, ψq| ď | covpπ`q˚µ, npφpNq, ψpNqq| ` 2 φα ψα e

´αN

ď C pφ8 ` φ1α e

αN q pψ8 ` ψ1α e

αN q e´κn ` 2 φα ψα e´αN

ď φα ψαpC e2αN´κn ` 2 e´αN q .

Taking N “ tκn4α u, C 1 “ C ` 2 eα and κ1 “ κ4 , we have

| covµ, npφ, ψq| ď C 1 φα ψα e´κ1n ,

and the result follows. l

In order to conclude Step 2, we now state the one-sided version of Theorem 9.2 and provehow it implies Theorem 9.2.6

Theorem 9.5. Let pΣ`, σ`q be a locally compact transitive one-sided topological Markov shiftwith alphabet A and transition matrix A, and let P` be a mixing σ`-invariant probabilitymeasure with full support on Σ`. Assume that

6Assumption (1) of Theorem 9.5 is far from being optimal, but will be sufficient for our purpose.

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(1) for every A-admissible finite sequence w “ pw0, . . . , wnq in A , the Jacobian of the mapfrom rwns to rws defined by pwn, x1, x2, . . . q ÞÑ pw0, . . . , wn, x1, x2, . . . q with respect tothe restrictions of the measure P` is constant;

(2) there exist a finite subset E of A and C 1, κ1 ą 0 such that for all n P N, we have

P``

tx P Σ` : x0 P E and τE pxq ě nu˘

ď C 1 e´κ1n . (9.3)

Then pΣ`,P`, σ`q has exponential decay of α-Hölder correlations.

Proof that Theorem 9.5 implies Theorem 9.2. Let pΣ, σ,P,E q be as in the statementof Theorem 9.2. Let P` “ pπ`q˚P, which is a mixing σ`-invariant probability measureon Σ`. Note that Hypothesis (1) in Theorem 9.5 follows from Hypothesis (1) of Theorem9.2. Similarly, Equation (9.3) follows from Equation (9.2). Hence Theorem 9.2 follows fromTheorem 9.5 and Proposition 9.4. l

Let us now consider Theorem 9.5. The scheme of its proof, using inducing and Youngtower arguments, was communicated to us by O. Sarig.

Step 3 : Proof of Theorem 9.5

In this final Step, using inducing of the dynamical system pΣ`, σ`q on the subspacetx P Σ` : x0 P E u “

Ť

aPE ras (a finite union of 1-cylinders), we present pΣ`, σ`q as a Youngtower to which we will apply the results of [You2].

Note that since σ` is mixing with full measure, there exists a σ`-invariant measurablesubset ∆ of Σ` such that the orbit under σ` of every element of ∆ passes infinitely manytimes inside the nonempty open subset

Ť

aPE ras. We again denote by τE : ∆ Ñ N ´ t0u therestriction to ∆ of the first return time in

Ť

aPE ras, so that if

∆0 “ tx P ∆ : x0 P E u “ď

aPE

∆X ras ,

then τE pxq “ mintn P N ´ t0u : σn`x P ∆0u for all x P ∆. We denote by F : ∆ Ñ ∆0 thefirst return map to ∆0 under iteration of the one-sided shift, that is

F : x ÞÑ στE pxq` pxq .

Let W be the set of admissible sequences w of length |w| at least 2 such that if w “

pw0, . . . , wnq thenw0, wn P E and w1, . . . , wn´1 R E .

We have the following properties:‚ the sets ∆a “ ∆X ras for a P E form a finite measurable partition of ∆0 and for every

a P E , the sets ∆w “ ∆X rws for w P W and w0 “ a form a countable measurable partitionof ∆a;‚ for every w PW , the first return time τE is constant (equal to |w|´1) on each ∆w, and

if w|w|´1 “ b, then the first return map F is a bijection from ∆w to ∆b;‚ for all w PW and x, y P ∆w, since x, y have the same |w| first components, we have

dpF pxq, F pyqq “ dpσ|w|´1` x, σ

|w|´1` yq “ e|w|´1 dpx, yq ě e dpx, yq ;

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‚ for all w PW , n P t0, . . . , |w| ´ 2u and x, y P ∆w, we have

dpσn`x, σn`yq “ en dpx, yq ď e|w|´2 dpx, yq ă dpF pxq, F pyqq ;

‚ for every w P W , the Jacobian of the first return map F : ∆w Ñ ∆w|w|´1for the

restrictions to ∆w and ∆w|w|´1of P` is constant.7

By an easy adaptation of [You2, Theo. 3] (see also [Mel1, §2.1]) which considers the casewhen E is a singleton, we have the following noneffective8 exponential decay of correlation:there exists κ ą 0 such that for every φ, ψ P C α

b pΣ`q, there exists a constant Cφ,ψ ą 0 suchthat

| covP`, npφ, ψq| ď Cφ,ψ e´κn

By an elegant argument using the Principle of Uniform Boundedness, it is proved in [ChCS,Appendix B] that this implies that there exists C, κ ą 0 such that for every φ, ψ P C α

b pΣ`q,we have

| covP`, npφ, ψq| ď C φα ψα e´κn .

This concludes the proof of Theorem 9.5, hence the proof of Theorem 9.1. l l

The next result gives examples of applications of Theorem 9.1 when ΓzX is infinite. Itstrengthens [AtGP, Theo. 2.1] that applies only to arithmetic lattices and only for the locallyconstant regularity (see Section 15.4), see also [BekL] for an approach using spectral gaps. Itwas claimed in [Kwo], but was retracted by the author.

Corollary 9.6. Let X be a locally finite simplicial tree without terminal vertices. Let Γ be ageometrically finite subgroup of AutpXq such that the smallest nonempty Γ-invariant subtreeof X is uniform without vertices of degree 2. Let α P s0, 1s.

(1) If LΓ “ Z, then the discrete time geodesic flow on ΓzGX has exponential decay ofα-Hölder correlations for the system of conductances c “ 0.

(2) If LΓ “ 2Z, then the square of the discrete time geodesic flow on ΓzGevenX has expo-nential decay of α-Hölder correlations for the 0 system of conductances, that is, there existC, κ ą 0 such that for all φ, ψ P C α

b pΓzGevenXq and n P Z, we have

ˇ

ˇ

ˇ

ż

ΓzGevenXφ ˝ g´2n ψ dmBM ´

1

mBMpΓzGevenXq

ż

ΓzGevenXφ dmBM

ż

ΓzGevenXψ dmBM

ˇ

ˇ

ˇ

ď C e´κ|n| φα ψα .

The main point of this corollary is to prove the exponential decay of volumes of geodesiclines going high in the cuspidal rays of ΓzX, stated as Assumption (9.1) in Theorem 9.1.There is a long history of similar results, starting from the exponential decay of volumesof small cusp neighbourhoods in noncompact finite volume hyperbolic manifolds (based onthe description of their ends) used by Sullivan to deduce Diophantine approximation results(see [Sul3, §9]).9 These results were extended to the case of locally symmetric Riemannian

7Actually, only a much weaker assumption is required, such as a Hölder-continuity property of this Jacobian,see [You2].

8Actually, there is in [You2] (see also [CyS]) a control on the constant in terms of some norms of the testfunctions, but these norms are not the ones we are interested in.

9and by probabilists in order to study the statistics of cusp excursions (see for instance [EF])

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manifolds by Kleinbock-Margulis [KM2] (based on the description of their ends using Siegelsets). Note that the geometrically finite lattice assumption on Γ is here in order to obtainsimilar descriptions of the ends of ΓzX.

Proof. Up to replacing X by C ΛΓ, we assume that X is a uniform simplicial tree withdegrees at least 3 and that Γ is a geometrically finite lattice of X. We use the 0 system ofconductances.

(1) By [Pau2] and as recalled in Section 2.7, the graph ΓzX is the union of a finite graph Yand finitely many geodesic rays Ri for i P t1, . . . , ku, such that if pxi, nqnPN is the sequence ofvertices in increasing order along Ri for i “ 1, . . . , k, then the vertex group Gxi, n of xi, n inthe quotient graph of groups ΓzzX satisfies Gxi, n Ă Gxi, n`1 for all n P N, and the edge groupof the edge ei,n with origin xi,n and endpoint xi, n`1 is equal to Gxi, n .10 Note that since thedegrees of X are at least 3, we have rGxi, n`1 : Gxi,ns ě 2 and |Gxi, 0 | ě 1, so that

|Gxi, n | ě 2n . (9.4)

Let E be the (finite) set of vertices V Y of Y. Note that for all n P N´ t0u and ` P ΓzGX,if `p0q P E and τEp`q ě 2n, then ` needs to leave Y after time 0 and it travels (geodesically)inside some cuspidal ray for a time at least n, so that there exists i P t1, . . . , ku such that`pnq “ xi, n. Hence for all n P N, using‚ the invariance of mBM under the (discrete time) geodesic flow in order to get the third

term,‚ Equation (8.4) where rxi, n is a fixed lift of xi, n in V X for the fifth term, and‚ Equation (9.4) since |Γ

rxi, n | “ |Gxi, n | and the fact that the degrees of the uniformsimplicial tree X are uniformly bounded and that the total mass of the Patterson measuresof the lattice Γ are uniformly bounded (see Proposition 4.14) for the last term,we have

mBM

`

t` P ΓzGX : `p0q P E and τEp`q ě 2nu˘

ď

kÿ

i“1

mBM

`

t` P ΓzGX : `pnq “ xi,nu˘

“

kÿ

i“1

mBM

`

t` P ΓzGX : `p0q “ xi,nu˘

“

kÿ

i“1

π˚mBMptxi,nuq

“

kÿ

i“1

1

|Γrxi, n |

ÿ

e, e1PEX : opeq“ope1q“ rxi, n, e‰e1

µrxi, npBeXq µrxi, npBe1Xq

ď k1

2nmaxxPV X

degpxq2 maxxPV X

µx2 .

The result then follows from Theorem 9.1 using the above finite set E which satisfiesAssumption (9.1) as we just proved, and using Propositions 4.14 and 4.15 in order to checkthat under the assumption that LΓ “ Z, the Bowen-Margulis measure mBM of Γ is finite andmixing under the discrete time geodesic flow on ΓzGX.

(2) The proof of Assertion (2) of Corollary 9.6 is similar to the one of Assertion (1). l

10identifying the edge group of an edge e with its image by the structural map Ge ÞÑ Gopeq

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Remark. The techniques introduced in the above proof in order to check the main hypothesisof Theorem 9.1 may be applied to numerous other examples. For instance, let X be a locallyfinite simplicial tree without terminal vertices. Let Γ be a nonelementary discrete subgroup ofAutpXq such that the smallest nonempty Γ-invariant subtree of X is uniform without verticesof degree 2, and such that LΓ “ Z. Let α P s0, 1s. Assume that ΓzX is the union of afinite graph A and finitely many trees T1, . . . ,Tk meeting A in one and exactly one vertex˚1, . . . , ˚n such that for every edge e in Ti pointing away from the root ˚i of Ti, the canonicalmorphism Ge Ñ Gopeq between edge and vertex groups of the quotient graph of groups ΓzzXis an isomorphism. Assume that there exists C, κ ą 0 such that for all n P N,

ÿ

i“1,...,k, xPV Ti : dpx,˚iq“n

1

|Gx|ď Ce´κn .

Then the discrete time geodesic flow on ΓzGX has exponential decay of α-Hölder correlationsfor the 0 system of conductances.

This is in particular the case for every k, q P N such that k ě 2, q ą 2k ` 1 and q ´ k isodd, when the quotient graph of groups ΓzzX has underlying edge-indexed graph11 a loop-edgewith both indices equal to q´k`1

2 glued to the root of a regular k-ary rooted tree, with indices1 for the edges pointing towards the root and q ´ k ` 1 for the edges pointing away from theroot (see the picture below with k “ 2). Note that X is then the pq ` 1q-regular tree, andthat the loop edge is here in order to ensure that LΓ “ Z. For instance, the vertex group ofa point at distance n from the root may be chosen to be Zp q´k`1

2 qZˆ`

Zpq ´ k ` 1qZ˘n.

q ´ 111

q ´ 1 q ´ 111

q ´ 1q ´ 111

q ´ 1q ´ 111

q ´ 1

q ´ 1q ´ 1

1

1

q´12

1

q ´ 1

1

q ´ 1

1

q ´ 1 q ´ 1

1

q´12

9.3 Rate of mixing for metric trees

Let pX, λq, X, Γ, rF , pµ˘x qxPV X and mF be as in the beginning of Section 4.4. The aim ofthis Section is to study the problem of finding conditions on these data under which the(continuous time) geodesic flow on ΓzGX is polynomially mixing for the Gibbs measure mF .

We will actually prove a stronger property, though it applies only to observables which aresmooth enough along the flow. Let us fix α P s0, 1s. Let pZ, µ, pφtqtPRq be a topological spaceZ endowed with a continuous one-parameter group pφtqtPR of homeomorphisms preservinga (Borel) probability measure µ on Z. For all k P N, let C k, α

b pZq be the real vector spaceof maps f : Z Ñ R such that for all z P Z, the map t ÞÑ fpφtxq is Ck-smooth, and such

11See definition in Section 2.7.

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that the maps Bitf : Z Ñ R defined by x ÞÑ di

dti |t“0fpφtxq for 0 ď i ď k are bounded and

α-Hölder-continuous. It is a Banach space when endowed with the norm

fk, α “kÿ

i“0

Bitfα ,

and it is contained in L2pZ, µq by the finiteness of µ. We denote by C k, αc pZq the vector

subspace of elements of C k, αb pZq with compact support.

For all ψ,ψ1 P L2pZ, µq and t P R, let

covµ, tpψ,ψ1q “

ż

Zψ ˝ φt ψ

1 dµ´

ż

Zψ dµ

ż

Zψ1 dµ

be the correlation coefficient of the observables ψ,ψ1 at time t under the flow pφtqtPR forthe measure µ. We say12 that the (continuous time) dynamical system pZ, µ, pφtqtPRq hassuperpolynomial decay of α-Hölder correlations if for every n P N there exist C “ Cn ą 0 andk “ kn P N such that for all ψ,ψ1 P C k, α

b pZq and t P R, we have

| covµ, tpψ,ψ1q| ď C p1` |t|q´n ψk, α ψ

1k, α .

Following Dolgopyat, we say that the dynamical system pZ, µ, pφtqtPRq is rapidly mixing if thereexists α ą 0 such that pZ, µ, pφtqtPRq has superpolynomial decay of α-Hölder correlations

The two assumptions on our data that we will use are the following ones, introducedrespectively in [Dol2] and [Mel1]. Recall that the Gibbs measure mF , when finite, is mixing ifand only the length spectrum LΓ is dense in R (see Theorem 4.8). The rapid mixing propertywill require stronger assumptions on LΓ.

We say that the length spectrum LΓ of Γ is 2-Diophantine if there exists a ratio of twotranslation lengths of elements of Γ which is Diophantine. Recall that a real number x isDiophantine if there exist α, β ą 0 such that

|x´p

q| ě α q´β

for all p, q P Z with q ą 0.Let E be a finite subset of vertices of ΓzX, and let rE be the set of vertices of X mapping

to E. We denote by TE the set of triples pλpγq, dpγq, qpγ, pqq where γ P Γ has translationlength λpγq ą 0, has dpγq vertices on its translation axis Axpγq modulo γZ and if the firstreturn time of a vertex p in rE X Axpγq in rE X Axpγq under the discrete time geodesic flowalong the translation axis has period qpγ, pq. We say that the length spectrum LΓ of Γ is4-Diophantine with respect to E if for all sequences pbkqkPN in r1`8r converging to `8 andpωkqkPN, pϕkqkPN in r0, 2πr , there exists N P N such that for all a ě N and C, β ě 1, thereexist k ě 1 and pτ, d, qq P TE such that

d`

pbkτ ` ωkdqtβ ln bku` qϕk, 2πZq ě C q b´ak .

We define the first return time after time ε on a finite subset E of vertices of ΓzX as themap τąεE : ΓzGX Ñ r0,`8s defined by τąεE p`q “ inftt ą ε : `ptq P Eu.

Theorem 9.7. Assume that the critical exponent δF is finite, that the Gibbs measure mF

is finite and mixing, and that the lengths of the edges of pX, λq have a finite upper bound.13

12See [Dol2], and more precisely [Mel1, Def. 2.2] whose definition is slightly different but implies the onegiven in this paper by the Principle of Uniform Boundedness argument of [ChCS, Appendix B] already usedin Section 9.2.

13They have a positive lower bound by definition, see Section 2.7.

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Furthermore assume that

(a) either ΓzX is compact and the length spectrum of ΓzX is 2-Diophantine,

(b) or there exists a finite subset E of vertices of ΓzX such that

(1) there exist C, κ ą 0 and ε P s0,minλr such that for all t ě 0,

mcpt` P ΓzGX : dp`p0q, Eq ď ε and τąεE p`q ě tuq ď C e´κ t ,

(2) the length spectrum of Γ is 4-Diophantine with respect to E.

Then the (continuous time) geodesic flow on ΓzGX has superpolynomial decay of α-Höldercorrelations for the normalised Gibbs measure mF

mF .

Note that the existence of E satisfying the exponentially small tail Hypothesis (1) is inparticular satisfied if Γ is geometrically finite with E the set of vertices of a finite subgraph ofΓzX whose complement in ΓzX is the underlying graph of a union of cuspidal rays in ΓzzX :see the proof of Corollary 9.6 and use the hypothesis on the lengths of edges.

Note that the exponentially small tail Hypothesis (1) might be weakened to a superpoly-nomially small tail hypothesis while keeping the same conclusion, see [Mel2]. Since the formeris easier to check than the latter, we prefer to state Theorem 9.7 as it is.

We will follow a scheme of proof analogous to the one in Section 9.2 for simplicial trees,by reducing the study to a problem of suspensions of Young towers, and then apply results of[Dol2] and [Mel1] for the rapid mixing property of suspensions of hyperbolic and nonuniformlyhyperbolic dynamical systems.

Proof. Since the Gibbs measure normalised to be a probability measure depends only on thecohomology class of the potential (see Equation (4.9)), we may assume by Proposition 3.12that F “ Fc is the potential on ΓzT 1X associated with a system of conductances rc : EXÑ Rfor Γ. We denote by δc the critical exponent of pΓ, Fcq, and by mc the Gibbs measure mFc .

Step 1 : Reduction to a suspension of a two-sided symbolic dynamics

We refer to the paragraphs before the statement of Theorem 5.9 for the definitions of‚ the system of conductances 7c for Γ on the simplicial tree X,‚ the (two-sided) topological Markov shift pΣ, σ,Pq on the alphabet A , conjugated to the

discrete time geodesic flow`

ΓzGX, 7g1,m7cm7c

˘

by the homeomorphism Θ : ΓzGXÑ Σ,

‚ the roof function r : Σ Ñ s0,`8r

‚ and the suspension pΣ, σ, aPqr “ pΣr, pσtrqtPR, aPrq over pΣ, σ, aPq with roof func-

tion r, where a “ 1Pr . This suspension is conjugated to the continuous time geodesic flow

`

ΓzGX, mcmc

, pgtqtPR˘

by the bilipschitz homeomorphism Θr : ΓzGX Ñ Σr defined at the endof the proof of Theorem 5.9. We will always (uniquely) represent the elements of Σr as rx, sswith x P Σ and 0 ď s ă rpxq.

Note that since Θ´1r conjugates pσtrqtPR and pgtqtPR, we have for all f : ΓzGX Ñ R and

x P Σr, when defined,

Bitpf ˝Θ´1r qpxq “

di

dti |t“0f ˝Θ´1

r pσtrxq “

di

dti |t“0fpgtΘ´1

r pxqq “ pBitfq ˝Θ´1

r pxq .

133 19/12/2016

Hence if f : ΓzGX Ñ R is Ck-smooth along the orbits of pgtqtPR, then f ˝Θ´1r is Ck-smooth

along the orbits of pσtrqtPR. Furthermore, since Θr is bilipschitz, the precomposition map byΘ´1r is a continuous linear isomorphism from C k, α

b pΓzGXq to C k, αb pΣrq.

Note that since Θr conjugates pgtqtPR and pσtrqtPR, and sends mcmc

to PrPr , we have, for all

ψ,ψ1 P L2pΓzGXq and t P R,

cov mcmc

, tpψ,ψ1q “ cov Pr

Pr, tpψ ˝Θ´1

r , ψ1 ˝Θ´1r q .

Therefore we only have to prove that the suspension pΣr, pσtrqtPR,

PrPrq is rapid mixing.

Step 2 : Reduction to a suspension of a one-sided symbolic dynamics

In this Step, we explain the rather standard reduction concerning mixing rates from sus-pensions of two-sided topological Markov shifts to suspensions of one-sided topological Markovshifts. We use the obvious modifications of the notation and constructions concerning the sus-pension of a noninvertible transformation to a semiflow, given for invertible transformationsat the beginning of Section 5.3.

We consider the one-sided topological Markov shift pΣ`, σ`,P`q over the alphabet Aconstructed at the beginning of Step 2 of the proof of Theorem 9.1, with the system ofconductances c now replaced by 7c. Let π` : Σ Ñ Σ` be the natural extension so thatP` “ pπ`q˚P and π` ˝ σ “ σ` ˝ π`.

We are going to construct in Step 2, as the suspension of pΣ`, σ`,P`q with an appro-priate roof function r`, a semiflow ppΣ`qr` ,

`

pσ`qtr`

˘

tě0, pP`qr`q, and prove that the flow

pΣr, pσtrqtPR,

PrPrq is rapid mixing if the semiflow ppΣ`qr` ,

`

pσ`qtr`

˘

tě0,pP`qr`pP`qr`

q is rapid mix-ing.

We start by introducing the notation that will be used in Step 2.

Let r` : Σ` Ñ s0,`8r be the map

r` : x ÞÑ λpe`0 q (9.5)

if x “ pxnqnPN P Σ` and x0 “ pe´0 , h0, e`0 q P A . Note that this map has a positive lower

bound, and a finite upper bound, and that it is locally constant (and even constant on the1-cylinders of Σ`). By Equation (5.10), we have

r` ˝ π` “ r . (9.6)

We denote by ppΣ`qr` ,`

pσ`qtr`

˘

tě0, pP`qr`q the suspension semiflow over pΣ`, σ`,P`q with

roof function r`. We (uniquely) represent the points of the suspension space pΣ`qr` as rx, ssfor x P Σ` and 0 ď s ă r`pxq. For all t ě 0, we have pσ`qtr`prx, ssq “ rσ

n`x, s

1s where n P Nand s1 P R are such that t` s “

řn´1i“0 r`pσ

i`xq ` s

1 and 0 ď s1 ď r`pσn`xq.

We define the suspended natural extension as the map π r` : Σr Ñ pΣ`qr` by

π r` : rx, ss ÞÑ rπ`pxq, ss ,

which is well defined by Equation (9.6). Note that π r` is 1-Lipschitz for the Bowen-Walters

distance on Σr and pΣ`qr` (see Proposition 5.11).

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For all ψ : Σr Ñ R and T ě 0, let us construct a function ψpT q : pΣ`qr` Ñ R as follows.For every rx, ss P pΣ`qr` , let N P N and s1 ě 0 be such that pσ`qTr`rx, ss “ rσ

N` x, s

1s, with

0 ď s1 ă r`pσN` xq and s` T “

N´1ÿ

i“0

r`pσi`xq ` s

1 .

LetψpT qprx, ssq “ ψpry, s1sq

where y “ pynqnPZ is such that yi “ xi`N if i ě ´N and yi “ zx0i`N otherwise. Note that

y0 “ xN , hence rpyq “ r`pσN` pxqq, and the above map is well defined.

Finally, for every ψ P C k, αb pΣrq or ψ P C k, α

b ppΣ`qr`q, let

ψk,8 “kÿ

i“0

Bitψ8

and

ψ1k, α “kÿ

i“0

Bitψ1α ,

so thatψk, α “ ψk,8 ` ψ

1k, α .

Lemma 9.8. Let T ě 0 and ψ P C k, αb pΣrq.

(1) For all t ě 0, we have pσ`qtr` ˝ πr` “ π r

` ˝ σtr.

(2) With α1 “ αsupλ , there exists a constant C1 ą 0 (independent of k, T and ψ) such that

|ψ ˝ σTr ´ ψpT q ˝ π r

`| ď C1 ψ1α e

´α1 T .

(3) We have ψpT q P C k, αb ppΣ`qr`q and ψpT q8 ď ψ8. With α2 “ α

inf λ , there exists aconstant C2 ą 0 (independent of k, T and ψ) such that

ψpT q1k, α ď C2 eα2 T ψ1k, α . (9.7)

Proof. (1) For every rx, ss P Σr, let n P N and s1 ě 0 be such that

t` s “n´1ÿ

i“0

r`pσi`π`pxqq ` s

1 and 0 ď s1 ď r`pσn`π`pxqq .

Since r` ˝ σi` ˝ π` “ r ˝ σi for all i P N, these two equalities are equivalent to

t` s “n´1ÿ

i“0

rpσixq ` s1 and 0 ď s1 ď rpσnxq .

Hencepσ`q

tr` ˝ π

r`prx, ssq “ pσ`q

tr`prπ`pxq, ssq “ rσ

n`π`pxq, s

1s

andπ r` ˝ σ

trprx, ssq “ π r

`prσnx, s1sq “ rπ`pσ

nxq, s1s .

135 19/12/2016

This proves Assertion (1) since π` ˝ σ “ σ` ˝ π`.

(2) By Proposition 5.11, we may assume that, in the formula of the Hölder norms, the Bowen-Walters distance is replaced by the function dBW, as this will only change C1 by CαBW C1.

For every rx, ss P Σr, with ry, s1s and N associated with π r`prx, ssq “ rπ`pxq, ss as in the

definition of ψpT qprπ`pxq, ssq, we have

dBWpσTr rx, ss, ry, s

1sq “ dBWprσNx, s1s, ry, s1sq ď dpσNx, yq ď e´N .

Since the positive roof function r is bounded from above by the least upper bound supλ ofthe lengths of the edges, we have

N ě

N´1ÿ

i“0

rpσixq

supλ“

1

supλps` T ´ s1q ě

T

supλ´ 1 .

Hence

|ψ ˝ σTr prx, ssq ´ ψpT qpπ r

`pxqq| “ |ψpσTr rx, ssq ´ ψpry, s

1sq|

ď ψ1α dBWpσTr rx, ss, ry, s

1sqα ď ψ1α e´ α

supλT`α

.

(3) The inequality ψpT q8 ď ψ8 is immediate by construction. Let us prove that ψpT q isCk along semiflow lines. Fix rx, ss P pΣ`qr` . Let us consider ε ą 0 small enough, so thatε ă r`pxq ´ s and ε ă r`pσ

N` xq ´ s

1, with ry, s1s and N as in the construction of ψpT qprx, ssq.Then

ψpT q ˝ pσ`qεr`prx, ssq “ ψpT qprx, s` εsq “ ψpry, s1 ` εsq “ ψ ˝ σεrpry, s

1sq .

Therefore by taking derivatives with respect to ε in this formula, ψpT q is indeed Ck alongsemiflow lines, and, for i “ 0, . . . , k, we have

Bit

`

ψpT q˘

“`

Bitψ˘pT q

. (9.8)

Let us prove that there exists a constant C2 ą 0 (independent of T and ψ) such that

ψpT q1α ď C2 eα2 T ψ1α . (9.9)

Let rx, ss P Σr, with ry, s1s and N as in the definition of ψpT qprx, ssq. Let rx, s s P Σr,with ry, s1s and N as in the definition of ψpT qprx, s sq. Up to exchanging rx, ss and rx, s s, weassume that N ď N .

By Proposition 5.11, we may assume that, in the Hölder norms formulas, the Bowen-Walters distance is replaced by the function dBW, as this will only change C2 by C2α

BW C2.Let

C3 “ minte´1, inf λu .

Note that the map dBW on Σr ˆ Σr is bounded from above by 1 ` supλ, since the distanceon Σ is at most 1 and since the roof function r is bounded from above by supλ.

We have

|ψpT qprx, ssq ´ ψpT qprx, s sq| “ |ψpry, s1sq ´ ψpry, s1sq| ď ψ1α dBWpry, s1s, ry, s1sqα . (9.10)

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If dBWprx, ss, rx, s sq ě C3 e´ T

inf λ , then

dBWpry, s1s, ry, s1sq ď 1` supλ ď

1` supλ

C3e

Tinf λ dBWprx, ss, rx, s sq .

Therefore Equation (9.9) follows from Equation (9.10) whenever C2 ěp1`supλqα

Cα3.

Conversely, suppose that dBWprx, ss, rx, s sq ă C3 e´ T

inf λ . Assume that

dBWprx, ss, rx, s sq “ dpx, xq ` |s´ s| ,

the other possibilities are treated similarly. Since

s` T “N´1ÿ

i“0

r`pσi`xq ` s

1

and since the roof function r` is bounded from below by inf λ, we have T ě N inf λ ´ inf λ,or equivalently N ď T

inf λ ` 1. Hence dpx, xq ă C3 e´ T

inf λ ď e´N by the definition of C3. Inparticular the sequences x and x indexed by N have the same N ` 1 first coefficients. Sincer`pzq depends only on z0 for all z P Σ`, we thus have r`pσi`xq “ r`pσ

i` xq for i “ 0, . . . , N .

Note that we have

s` T “

N´1ÿ

i“0

r`pσi` xq ` s

1 .

If N “ N , then by taking the difference of the last two centred equations, we haves ´ s “ s1 ´ s1, and by construction, the sequences y and y indexed by Z satisfy yi “ pyqi ifi ď 0 and if 0 ď i ď ´ ln dpx, xq ´N . Therefore dpy, yq ď eN dpx, xq and

dBWpry, s1s, ry, s1sq ď dpy, yq ` |s1 ´ s1| ď eN dpx, xq ` |s´ s|

ď eN dBWprx, ss, rx, s sq ď eT

inf λ`1 dBWprx, ss, rx, s sq .

Therefore Equation (9.9) follows from Equation (9.10) whenever C2 ě eα.If on the contrary N ą N , then again by difference

s´ s “

N´1ÿ

i“N`1

r`pσi` xq ` r`pσ

N` xq ´ s

1 ` s1.

Note that s1 ě 0, that r`pσN` xq´s1 ě 0, and that |s´s| ă C3 e´ T

inf λ ď inf λ by the definitionof C3. Hence we have N “ N`1 and s´s “ r`pσ

N` xq´s

1`s1. By construction, the sequencesσy and y indexed by Z satisfy pσyqi “ pyqi if i ď 0 and if 0 ď i ď ´ ln dpx, xq ´N ´ 1. Henceby the definition of dBW and since rpyq “ r`pσ

N` xq, we have

dBWpry, s1s, ry, s1sq ď dpσy, yq ` rpyq ´ s1 ` s1 ď eN`1 dpx, xq ` |s´ s|

ď eN`1 dBWprx, ss, rx, s sq ď eT

inf λ`2 dBWprx, ss, rx, s sq .

Therefore Equation (9.9) follows from Equation (9.10) whenever C2 ě e2α. This ends theproof of Equation (9.9).

Now note that Equations (9.8) and (9.9) imply Equation (9.7) by summation (using theindependence of C2 on ψ), thus concluding the proof of Lemma 9.8. l

137 19/12/2016

Proposition 9.9. Let µ be a pσtrqtPR-invariant probability measure on Σr. Assume that thedynamical system ppΣ`qr` , ppσ`q

tr`qtPR, pπ

r`q˚µq has superpolynomial decay of α-Hölder cor-

relations. Then pΣr, pσtrqtPR, µq has superpolynomial decay of α-Hölder correlations.

Proof. We fix n P N. Let N “ 1` 2rsupλinf λ s. Let k P N and C4 ą 0 (depending on n) be such

that for all ψ,ψ1 P C k, αb ppΣ`qr`q, we have for all t ě 1

| covpπ r`q˚µ, tpψ,ψ1q| ď C4 ψk, α ψ

1k, α tN n . (9.11)

Now let ψ,ψ1 P C k, αb pΣrq. We again denote by ˘ t any value in r´t, ts for any t ě 0. By

invariance of µ under pσtrqtPR, by Lemma 9.8 (2) and by Lemma 9.8 (1), we have, for anyT ě 0 (to be chosen appropriately later on),

ż

Σr

ψ ˝ σtr ψ1 dµ “

ż

Σr

ψ ˝ σT`tr ψ1 ˝ σTr dµ

“

ż

Σr

pψpT q ˝ π r` ˘ C1 ψ

1α e

´α1 T q ˝ σtr pψ1pT q ˝ π r

` ˘ C1 ψ11α e

´α1 T q dµ

“

ż

pΣ`qr`

ψpT q ˝ pσ`qtr` ψ1

pT qdpπ r

`q˚µ ˘ C21 ψα ψ

1α e´α1 T .

A similar estimate holds for the second term in the definition of the correlation coefficients.Hence, applying Equation (9.11) to the observables ψpT q and ψ1pT q, by Lemma 9.8 (3), wehave

| covµ, tpψ,ψ1q| ď | covpπ r`q˚µ, tpψ

pT q, ψ1pT qq| ` 2 C2

1 ψα ψ1α e

´α1 T

ď C4 pψk,8 ` C2 ψ1k, α e

α2 T q pψ1k,8 ` C2 ψ11k, α e

α2 T q t´N n

` 2 C21 ψα ψ

1α e´α1 T

ď ψk, α ψ1k, α pC4 C

22 e

2α2T t´N n ` 2 C21 e

´α1 T q .

Take T “ nα1 ln t ě 0. Since N “ 1 ` 2rα

2

α1 s, we have 2α2 nα1 ´ N n ď ´n. Hence with

C5 “ C4 C22 ` 2 C2

1 , we have for all t ě 1

| covµ, tpψ,ψ1q| ď C5 ψk, α ψ

1k, α t´n .

This concludes the proof of Proposition 9.9. l

Step 3 : Conclusion of the proof of Theorem 9.7

In this Step, we prove that the semiflow ppΣ`qr` ,`

pσ`qtr`

˘

tě0,pP`qr`pP`qr`

q is rapid mixing,

which concludes the proof of Theorem 9.7, using Proposition 9.9 with µ “ PrPr .

Recall (see the proof of Theorem 5.9) that Y “ t` P ΓzGX : `p0q P V Xu is a cross-sectionof the geodesic flow on ΓzGX, and that if R : Y Ñ ΓzGX is the reparametrisation mapof ` P Y to a discrete geodesic line 7` P ΓzGX with the same origin, then the measure µY ,induced by the Gibbs measure mc on the cross-section Y by disintegration along the flow,maps by R˚ to a constant multiple of m7c (see Lemma 5.10 (2)). Hence for all n P N, by

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Assumption (b) (1) in the statement of Theorem 9.7, we have

m7c

´!

7` P ΓzGX :7`p0q P E

@ k P t1, . . . , n´ 1u, 7`pkq R E

)¯

ďm7c

µY µY

´!

R´1p7`q P Y :R´1p7`qp0q P E

@ t P s0, n inf λr , R´1p7`qptq R E

)¯

ďm7c

ε µY mc

´!

gsR´1p7`q P ΓzGX :0 ď s ď ε, dpgsR´1p7`qp0q, Eq ď ε@ t P sε, n inf λ´ εr , gsR´1p7`qptq R E

)¯

ďm7c

ε µY C e´κ pinf λqn`κ ε .

Therefore Equation (9.1) (where c is replaced by 7c) is satisfied, with C 1 “m7cC e

κ ε

ε µY and

κ1 “ κ inf λ. As seen in the proof of Theorem 9.1, this implies that there exists a finite subsetE of the alphabet A such that Equation (9.3) is satisfied.

We now apply [Mel1, Theo. 2.3] with the dynamical system pX,m0, T q “ pΣ`,P`, σ`q(using the system of conductances 7c) and the roof function h “ r`. This dynamical systemis presented as a Young tower in Step 3 of the proof of Theorem 9.1. Equation (9.3) forthe first return map τE and the 4-Diophantine hypothesis are exactly the hypothesis neededin order to apply [Mel1, Theo. 2.3]. Thus the semiflow ppΣ`qr` ,

`

pσ`qtr`

˘

tě0,pP`qr`pP`qr`

q hassuperpolynomial decay of α-Hölder correlations.

When ΓzX is compact, the alphabet A is finite and pΣ`, σ`,P`q is a (one-sided) subshiftof finite type, hence we do not need the exponentially small tail assumption, and only the2-Diophantine hypothesis, and we may apply [Dol2]. l

Corollary 9.10. Assume that the critical exponent δF is finite, that the Gibbs measure mF

is finite and mixing, that the lengths of the edges of pX, λq have a finite upper bound, andthat ΓzX is geometrically finite. There exists a full measure subset A of R4 (for the Lebesguemeasure) such that if Γ has a quadruple of translation lengths in A, then the (continuous time)geodesic flow on ΓzGX has superpolynomial decay of α-Hölder correlations for the Bowen-Margulis measure mBM.

Proof. The exponentially small tail Assumption (b) (1) is checked as in the proof of Corollary9.6. The deduction of Corollary 9.10 from Theorem 9.7 then proceeds, by an argument goingback in part to Dolgopyat, as for the deduction of Corollary 2.4 from Theorem 2.3 in [Mel1].l

Note that under the general assumptions of Theorem 4.8, the geodesic flow on ΓzGXmight not be exponentially mixing, see for instance [Pol1, page 162] or [Rue2] for analogousbehaviour.

139 19/12/2016

140 19/12/2016

Part II

Geometric equidistribution andcounting

141 19/12/2016

Chapter 10

Equidistribution of equidistant levelsets to Gibbs measures

Let X be a geodesically complete proper CATp´1q space, let Γ be a nonelementary discretegroup of isometries of X, let rF be a continuous Γ-invariant map on T 1X such that δ “ δΓ, F˘

is finite and positive and that the triple pX,Γ, rF q satisfies the HC-property,1 and let pµ˘x qxPXbe Patterson densities for the pairs pΓ, F˘q.

In this Chapter, we prove that the skinning measure on (any nontrivial piece of) the outerunit normal bundle of any properly immersed nonempty proper closed convex subset of X,pushed a long time by the geodesic flow, equidistributes towards the Gibbs measure, underfiniteness and mixing assumptions. This result gives four important extensions of [PaP14a,Thm. 1], one for general CATp´1q spaces with constant potentials, one for Riemannian man-ifolds with pinched negative curvature and Hölder potentials, one for R-trees with generalpotentials, and one for simplicial trees.

10.1 A general equidistribution result

Before stating this equidistribution result, we start by a technical construction which will alsobe useful in the following Chapter 11. We refer to Section 2.5 for the notation concerning thedynamical neighbourhoods (including V ¯w, η1, R) and to Chapter 7 for the notation concerningthe skinning measures (including ν˘w ).

Technical construction of bump functions. Let D˘ be nonempty proper closed convexsubsets of X, and let R ą 0 be such that ν˘w pV

¯w, η1, Rq ą 0 for all η1 ą 0 and w P B1

¯D˘. Let

η ą 0 and let Ω˘ be measurable subsets of B1¯D

˘. We now construct functions φ¯η,R,Ω˘

:

GX Ñ r0,`8r whose supports are contained in dynamical neigbourhoods of Ω˘. If X “ ĂM isa manifold and rF “ 0, we recover the same bump functions after the standard identifications.

For all η1 ą 0, let h˘η, η1 : G¯X Ñ r0,`8r be the Γ-invariant measurable maps defined by

h˘η, η1pwq “1

ν˘w pV¯w, η, η1q

(10.1)

if ν˘w pV¯w, η, η1q ą 0 (which is for instance satisfied if w˘ P ΛΓ) and h˘η, η1pwq “ 0 otherwise.

1See Definition 3.4.

143 19/12/2016

These functions h˘η, η1 have the following behaviour under precomposition by the geodesicflow. By Lemma 7.5 (2), by Equation (2.13), and by the invariance of ν˘w under the geodesicflow, we have, for all t P R and w P G˘X,

h¯η, η1pg¯twq “ eC

˘w˘pwp0q, wp¯tqq h¯

η, e´tη1pwq . (10.2)

Let us also describe the behaviour of h˘η, η1 when η1 is small. Let w P G˘X be such that w

is isometric at least on ˘r0,`8r, which is for instance the case if w P B1˘D

¯. For all η1 ą 0and ` P B˘pw, η1q, let pw be an extension of w such that dW˘pwqp`, pwq ă η1. Then pwp0q “ wp0qby the assumption on w, and using [PauPS, Lem. 2.4],2 we have

dp`p0q, wp0qq “ dp`p0q, pwp0qq ď dW˘pwqp`, pwq ă η1 .

Hence, with κ1 and κ2 the constants in the Definition 3.4, if

c1 “ κ1 ` δ ` supπ´1pBpwp0q, η1qq

| rF | ,

we have, by Proposition 3.10 (2),

| C¯w¯pwp0q, `p0qq | ď c1 pη1qκ2 .

Using the defining Equation (7.12) of ν¯w , for all s P R, η1 ą 0 and ` P B˘pw, η1q, we have

e´c1pη1qκ2

ds dµW˘pwqp`q ď dν¯w pgs`q ď ec1pη

1qκ2ds dµW˘pwqp`q .

It follows that for all η1 P s0, 1s and w P B1˘D

¯ such that w˘ P ΛΓ, we have the followingcontrol of h¯η, η1pwq:

e´c1pη1qκ2

2η µW˘pwqpB˘pw, η1qq

ď h¯η, η1pwq ďec1pη

1qκ2

2η µW˘pwqpB˘pw, η1qq

. (10.3)

Note that when X is an R-tree, we may take κ2 “ 1 and c1 “ rF ´ δ8 in this equation, aswe saw in the proof of Proposition 3.5.

Recall that 1A denotes the characteristic function of a subset A. We now define the testfunctions φ¯

η,R,Ω˘: GX Ñ r0,`8r by

φ¯η,R,Ω¯

“ h¯η,R ˝ f˘

D¯1V ˘η,RpΩ

¯q, (10.4)

where V ˘η,RpΩ

¯q and f˘D¯

are as in Section 2.5. Note that if ` P V ˘η,RpΩ

¯q, then `˘ R B8D¯

by convexity. Thus, ` belongs to the domain of definition U ˘

D¯of f˘

D¯; hence φ¯

η,R,Ω¯p`q “

h¯η,R ˝ f˘

D¯p`q is well defined. By convention, φ¯

η,R,Ω¯p`q “ 0 if ` R V ˘

η,RpΩ¯q.

The following property of the bump functions is proved as in [PaP14a, Prop. 18], usingthe disintegration result of Proposition 7.6.

Lemma 10.1. For every η ą 0, the functions φ¯η,R,Ω˘

are measurable, nonnegative and satisfyż

GXφ¯η,R,Ω˘

drmF “ rσ˘pΩ¯q . l

2Although it is stated for Riemannian manifolds, the argument is valid in general CATp´1q spaces.

144 19/12/2016

We now state and prove the aforementioned equidistribution result. Note that as elementsof the outer unit normal bundles are now only geodesic rays on r0,`8r, their pushforwardsby the geodesic flow at time t are geodesic rays on r´t,`8r and the convergence towardsgeodesic lines (defined on s´8,`8r) does take place in the full space of generalised geodesiclines

p

GX. This explains why it is important not to forget to consider the negative times –skinning measures, supported on geodesic rays, pushed by the geodesic flow, have a chanceto weak-star converge to Gibbs measures, supported on geodesic lines, up to renormalisation.

The proof of the following result has similarities with that of [PaP14a, Theo. 1], but thecomputations do not apply in the present context because the proof in loc. cit. does not keeptrack of the past: here we can no longer reduce our study to the outer unit normal bundle ofthe t-neighbourhood of the elements of D .

Theorem 10.2. Let pX,Γ, rF q be as in the beginning of Chapter 10. Assume that the Gibbsmeasure mF on ΓzGX is finite and mixing for the geodesic flow. Let D “ pDiqiPI be a locallyfinite Γ-equivariant family of nonempty proper closed convex subsets of X with ΓzI finite.Let Ω “ pΩiqiPI be a locally finite Γ-equivariant family of measurable subsets of

p

GX, withΩi Ă B

1˘Di for all i P I. Assume that σ˘Ω is finite and nonzero. Then, as t Ñ `8, for the

weak-star convergence of measures on Γz

p

GX,

1

pg˘tq˚σ˘Ωpg˘tq˚σ

˘Ω

˚á

1

mF mF .

Proof. We only give the proof when ˘ “ `, the other case is treated similarly. Given threenumbers a, b, c (depending on some parameters), we write a “ b˘ c if |a´ b| ď c.

Let η P s0, 1s. As in the proof of [PaP14a, Thm. 19], we may assume that ΓzI is finite.Hence, using Lemma 7.5 (i), we may fix R ą 0 such that ν´w pV

`w, η,Rq ą 0 for all w P B1

`Di

and i P I.Using the notation introduced in the above construction of the bump functions φ´η,R,Ωi ,

we may hence consider the global test functions rΦη : GX Ñ r0,`8r,

rΦηpvq “ÿ

iPI„

φ´η,R,Ωi “ÿ

iPI„

h´η,R ˝ f`Di1V `η,RpΩiq

.

As in loc. cit., the function rΦη : GX Ñ r0,`8r is well defined (independent of the represen-tatives of i), measurable and Γ-equivariant. Hence it defines, by passing to the quotient, ameasurable function Φη : ΓzGX Ñ r0,`8r . By Lemma 10.1, the function Φη is integrableand satisfies

ż

ΓzGXΦη dmF “ σ

`Ω . (10.5)

Fix ψ P CcpΓz

p

GXq. Let us prove that

limtÑ`8

1

pgtq˚σ`Ω

ż

ΓzGXψ dpgtq˚σ

`Ω “

1

mF

ż

ΓzGXψ dmF .

Consider a fundamental domain ∆Γ for the action of Γ onp

GX, such that the boundary of∆Γ has zero measure, the interiors of its translates are disjoint and any compact subset ofp

GX meets only finitely many translates of ∆Γ: see [Rob2, p. 13] (or the proof of [PaP14a,

145 19/12/2016

Prop. 18]), using the fact that rmF has no atoms, since mF is finite and according to Corollary4.6(1) and to Theorem 4.5.

By a standard argument of finite partition of unity and up to modifying ∆Γ, we mayassume that there exists a function rψ :

p

GX Ñ R whose support has a small neighbourhoodcontained in ∆Γ such that rψ “ ψ ˝ p, where p :

p

GX Ñ Γz

p

GX is the canonical projection(which is Lipschitz). Fix ε ą 0. Since rψ is uniformly continuous, for every η ą 0 small enoughand for every t ě 0 large enough, for all w P G`X and ` P V `

w, η, e´tR, we have

rψp`q “ rψpwq ˘ε

2. (10.6)

If t is big enough and η small enough, we have, using respectively• the definition of the global test function rΦη, since the support of rψ is contained in ∆Γ and

the support of φ´η,R,Ωi is contained in U `Di, for the second equality,

• the disintegration property of f`Di in Proposition 7.6 for the third equality,• the fact that if ` is in the support of ν´ρ , then f

`Dipg´t`q “ f`Dip`q “ ρ and the change of

variables by the geodesic flow w “ gtρ for the fourth equality,• the fact that the support of ν´

g´twis contained in W 0`pg´twq, and that

W 0`pg´twq X gtV `η,RpΩiq “ gt`

W 0`pg´twq X V `η,RpΩiqq “ gtV `g´tw, η,R

“ V `w, η, e´tR

for the fifth equality,• Equation (10.6) for the sixth equality, and• the definition of h´, the invariance of the measure ν´

g´twand the Gibbs measure rmF under

the geodesic flow, and the definition of the measure σ`Ω for the last two equalities.ż

ΓzGXψ Φη ˝ g

´t dmF “

ż

∆ΓXGX

rψ rΦη ˝ g´t drmF

“ÿ

iPI„

ż

`PU `Di

rψp`q φ´η,R,Ωipg´t`q drmF p`q

“ÿ

iPI„

ż

ρPB1`Di

ż

`PU `Di

rψp`q h´η,Rpf`Dipg´t`qq1V `η,RpΩiq

pg´t`q dν´ρ p`q drσ`Dipρq

“ÿ

iPI„

ż

wPgtB1`Di

ż

`PgtV `η,RpΩiq

rψp`q h´η,Rpg´twqq dν´

g´twp`q dpgtq˚rσ

`Dipwq

“ÿ

iPI„

ż

wPgtB1`Di

ż

`PV `w, η, e´tR

rψp`q h´η,Rpg´twq dν´

g´twp`q dpgtq˚rσ

`Dipwq

“ÿ

iPI„

ż

wPgtB1`Di

rψpwq h´η,Rpg´twq ν´

g´twpgtV `

g´tw, η,Rq dpgtq˚rσ

`Dipwq

˘ε

2

ż

∆ΓXGX

rΦη ˝ g´t drmF

“ÿ

iPI„

ż

p

GX

rψ dpgtq˚rσ`Di˘ε

2

ż

ΓzGXΦη ˝ g

´t dmF

“

ż

Γz

p

GXψ dpgtq˚σ

`Ω ˘

ε

2

ż

ΓzGXΦη dmF (10.7)

146 19/12/2016

We then conclude as in the end of the proof of [PaP14a, Thm. 19]. By Equation (10.5), wehave pgtq˚σ`Ω “ σ`Ω “

ş

ΓzGX Φη dmF . By the mixing property of the geodesic flow onΓzGX for the Gibbs measure mF , for t ě 0 large enough (while η is small but fixed), wehence have

ş

Γz

p

GXψ dpgtq˚σ

`Ω

pgtq˚σ`Ω

“

ş

ΓzGX Φη ˝ g´t ψ dmF

ş

ΓzGX Φη dmF˘ε

2“

ş

ΓzGX ψ dmF

mF ˘ ε .

This proves the result. l

Recall that by Proposition 3.5, Theorem 10.2 applies to Riemannian manifolds withpinched negative curvature and for R-trees for which the geodesic flow is mixing and whichsatisfy the finiteness requirements of the Theorem.

Since pushforwards of measures are weak-star continuous and preserve total mass, wehave, under the assumptions of Theorem 10.2, the following equidistribution result in X ofthe immersed t-neighbourhood of a properly immersed nonempty proper closed convex subsetof X: as tÑ `8,

1

σ`Ωπ˚pg

tq˚σ`Ω

˚á

1

mF π˚mF . (10.8)

10.2 Rate of equidistribution of equidistant level sets for man-ifolds

If X “ ĂM is a simply connected Riemannian manifold of pinched negative curvature and ifthe geodesic flow of ΓzĂM is mixing with exponentially decaying correlations, we get a versionof Theorem 10.2 with error bounds. See Section 9.1 for conditions on Γ and rF that implyexponential mixing.

Theorem 10.3. Let ĂM be a complete simply connected Riemannian manifold with negativesectional curvature. Let Γ be a nonelementary discrete group of isometries of ĂM . Let rF :T 1

ĂM Ñ R be a bounded Γ-invariant Hölder-continuous function with positive critical exponentδ “ δΓ, F . Let D “ pDiqiPI be a locally finite Γ-equivariant family of nonempty proper closedconvex subsets of ĂM , with finite nonzero skinning measure σD . Let M “ ΓzĂM and let F :T 1M Ñ R be the potential induced by rF .(i) If M is compact and if the geodesic flow on T 1M is mixing with exponential speed for theHölder regularity for the potential F , then there exist α P s0, 1s and κ2 ą 0 such that for allψ P C α

c pT1Mq, we have, as tÑ `8,

1

σD

ż

ψ dpgtq˚σD “1

mF

ż

ψ dmF `Ope´κ2t ψαq .

(ii) If ĂM is a symmetric space, if Di has smooth boundary for every i P I, if mF is finiteand smooth, and if the geodesic flow on T 1M is mixing with exponential speed for the Sobolevregularity for the potential F , then there exists ` P N and κ2 ą 0 such that for all ψ P

C `c pT

1Mq, we have, as tÑ `8,

1

σD

ż

ψ dpgtq˚σD “1

mF

ż

ψ dmF `Ope´κ2t ψ`q .

147 19/12/2016

Note that if ĂM is a symmetric space and M has finite volume, then M is geometricallyfinite. Theorem 4.7 implies that mF is finite if F is small enough. The maps Op¨q depend onĂM,Γ, F,D , and the speeds of mixing.

Proof. Up to rescaling, we may assume that the sectional curvature is bounded from aboveby ´1. The critical exponent δ and the Gibbs measure mF are finite in all the cases weconsider here.

The deduction of this result from the proof of Theorem 10.2 by regularisations of theglobal test function Φη introduced in the proof of Theorem 10.2 is analogous to the deductionof [PaP14a, Theo. 20] from [PaP14a, Theo. 19] when F “ 0. The doubling property of thePatterson densities and the Gibbs measure for general F , required by this deduction in theHölder regularity case, is given by [PauPS, Prop. 3.12]. For the assertion (ii), the requiredsmoothness of mF (that is, the fact that mF is absolutely continuous with respect to theLebesgue measure with smooth Radon-Nikodym derivative) allows to use the convolutionapproximation. l

10.3 Equidistribution of equidistant level sets on simplicialgraphs and random walks on graphs of groups

Let X, X,Γ,rc,ĂFc, δc be as in the beginning of Section 9.2. Let rFc : T 1X Ñ R be its associatedpotential, and let δ “ δc be the critical exponent of c. Let pµ˘x qxPV X be two Patterson densitieson B8X for the pairs pΓ, F˘c q, and let mc “ mFc be the associated Gibbs measure on ΓzGX.

In this Section, we state an equidistribution result analogous to Theorem 10.2, which nowholds in the space of generalised discrete geodesic lines Γz

p

GX, but whose proof is completelyanalogous.

Theorem 10.4. Let X,Γ,rc, pµ˘x qxPV X be as above, with δc finite. Assume that the Gibbsmeasure mc on ΓzGX is finite and mixing for the discrete time geodesic flow. Let D “ pDiqiPI

be a locally finite Γ-equivariant family of nonempty proper simplicial subtrees of X. Let Ω “

pΩiqiPI be a locally finite Γ-equivariant family of measurable subsets ofp

GX, with Ωi Ă B1`Di

for all i P I. Assume that σ`Ω is finite and nonzero. Then, as n Ñ `8, for the weak-starconvergence of measures on Γz

p

GX,

1

pgnq˚σ`Ω

π˚pgnq˚σ

`Ω

˚á

1

mF mF . l

We leave to the reader the analog of this result when the restriction to ΓzGevenX of theGibbs measure is finite and mixing for the square of the discrete time geodesic flow.

Using Propositions 4.14 and 4.15 in order to check that the Bowen-Margulis measure mBM

on ΓzGX is finite and mixing, we have the following consequence of Theorem 10.4, using thesystem of conductances rc “ 0.

Corollary 10.5. Let X be a uniform simplicial tree. Let Γ be a lattice of X such that the graphΓzX is not bipartite. Let D “ pDiqiPI be a locally finite Γ-equivariant family of nonemptyproper simplicial subtrees of X. Let Ω “ pΩiqiPI be a locally finite Γ-equivariant family ofmeasurable subsets of

p

GX, with Ωi Ă B1`Di for all i P I. Assume that the skinning measure

148 19/12/2016

σ`Ω (with vanishing potential) is finite and nonzero. Then, as n Ñ `8, for the weak-starconvergence of measures on Γz

p

GX,

1

pgnq˚σ`Ω

π˚pgnq˚σ

`Ω

˚á

1

mBMmBM . l

When furthermore X is regular, we have the following corollary, using Proposition 8.1 (3).

Corollary 10.6. Let X be a regular simplicial tree of degree at least 3. Let Γ be a lattice ofX such that the graph ΓzX is not bipartite. Let D “ pDiqiPI be a locally finite Γ-equivariantfamily of nonempty proper simplicial subtrees of X. Let Ω “ pΩiqiPI be a locally finite Γ-equivariant family of measurable subsets of

p

GX, with Ωi Ă B1`Di for all i P I. Assume that

the skinning measure σ`Ω (with vanishing potential) is finite and nonzero. Then, as nÑ `8,for the weak-star convergence of measures on Γz

p

GX,

1

pgnq˚σ`Ω

π˚pgnq˚σ

`Ω

˚á

1

VolpΓzzXqvolΓzzX . l

Let us give an application of Corollary 10.6 in terms of random walks on graphs of groups,which might also be deduced from general result on random walks, as indicated by M. Burgerand S. Mozes.

Let pY, G˚q be a connected graph of finite groups with finite volume, and let pY1, G1˚q bea connected subgraph of subgroups.3 Note that pY1, G1˚q also has finite volume, less than orequal to the volume of pY, G˚q. We say that pY, G˚q is homogeneous if

ř

ePEY, opeq“x|Gx||Ge|

isconstant at least 3 for all x P V Y. We say that a connected graph of groups is 2-acylindrical ifthe action of its fundamental group on its Bass-Serre tree is 2-acylindrical (see Remark 5.4).In particular, this action is faithful if the graph has at least two edges.

The non-backtracking simple random walk on pY, G˚q starting transversally to pY1, G1˚qis the following Markovian random process pXn “ pfn, γnqqnPN where fn P EY and γn PGopfnq for all n P N. Choose at random a vertex y0 of Y1 for the probability measure

1VolpY1,G1˚q

volY1,G1˚ (we will call y0 the origin of the random path). Then choose uniformlyat random X0 “ pf0, γ0q where f0 P EY is such that opf0q “ y0 and γ0 is a double coset inG1y0

zGy0ρ f0pGf0q such that if f0 P EY1 then γ0 R G

1y0ρ f0pGf0q.4 Assuming Xn “ pfn, γnq

constructed, choose uniformly at random Xn`1 “ pfn`1, γn`1q where fn`1 P EY is suchthat opfn`1q “ tpfnq and γn`1 P Gopfn`1qρ fn`1

pGfn`1q is such that if fn`1 “ fn thenγn`1 R ρ fn`1

pGfn`1q. The n-th vertex of pXn “ pfn, γnqqnPN is opfnq.

Corollary 10.7. Let pY, G˚q be a homogeneous 2-acylindrical nonbipartite connected graph offinite groups with finite volume, and let pY1, G1˚q be a homogeneous nonempty proper connectedsubgraph of subgroups. Then the n-th vertex of the non-backtracking simple random walk onpY, G˚q starting transversally to pY1, G1˚q converges in distribution to 1

VolpY,G˚q volY,G˚ asnÑ `8.

Proof. Let Γ be the fundamental group of pY, G˚q (with respect to a choice of basepointin V Y1), which is a lattice of the Bass-Serre tree X of pY, G˚q, since Γ acts faithfully on X

3See Section 2.7 for definitions and background.4This last condition says that γ0 is not the double coset of the trivial element.

149 19/12/2016

and pY, G˚q has finite volume. Note that X is regular since pY, G˚q is homogeneous. Letp : XÑ Y “ ΓzX be the canonical projection.

Let Γ1 be the fundamental group of pY1, G1˚q (with respect to the same choice of basepoint).By [Bass, 2.15], there exists a simplicial subtree X1 whose stabiliser in Γ is Γ1, such that thequotient graph of groups Γ1zzX1 identifies with pY1, G1˚q. Similarly, X1 is regular since pY1, G1˚qis homogeneous. Let D “ pγX1qγPΓΓ1 , which is a locally finite family of nonempty propersimplicial subtrees of X.

Using the notation of Example 2.10 for the graph of groups ΓzzX (which identifies withpY, G˚q), we fix lifts rf and ry in X by p of every edge f and vertex y of Y such that rf “ rf ,and elements gf P Γ such that gf Ątpfq “ tp rfq. We may assume that rf P EX1 if f P EY1, thatry P V X1 if y P V Y1, and that gf P Γ1 if f P EY1.

Let pΩ,Pq be the (canonically constructed) probability space of the random walk pXn “

pfn, γnqqnPN. For all n P N, let yn “ opfnq be the random variable (with values in the discretespace Y “ ΓzX) of the n-th vertex of the random walk pXnqnPN.

Let us define a measurable map Θ : Ω Ñ Γz

p

GX, with image contained in the image ofB1`X1 by the canonical projection

p

GX Ñ Γz

p

GX, such that Θ˚P is the normalised skinning

measure σ`Dσ`D

and that the following diagram commutes for all n P N :

ΩΘ

ÝÝÝÝÑ Γz

p

GXynŒ Öπ˝gn

Y “ ΓzX . (10.9)

Assuming that we have such a map, we have

pynq˚P “ pπ˚ ˝ pgnq˚ ˝Θ˚qP “ π˚pg

nq˚σ`Dσ`D

“1

pgnq˚σ`D

π˚pgnq˚σ

`D

so that the convergence of the law of yn to 1VolpY,G˚q volY,G˚ follows from Corollary 10.6

applied to B1`D .

Y

X

p

fn fn`1yn`1yn yn`2

αn

Ăynγn

αn`1

Ćyn`1

en en`1

g ´1fn

Ăfn

g ´1fn

Ăfn

γn`1g ´1fn`1

Ćfn`1

g ´1fn

g fn

y0 f0

ry0

rf0

g f0

e0

g ´1f0

rf0 γ0

150 19/12/2016

Let pXn “ pfn, γnqqnPN be a random path with origin y0 P Y1, corresponding to ω P Ω.Fix a representative of γn in its right class for every n ě 1, and a representative of γ0 in itsdouble class, that we still denote by γn and γ0 respectively. Using ideas used for the codingintroduced in Section 5.2, let us construct by induction an infinite geodesic edge path penqnPNwith origin ope0q “ ry0 and a sequence pαnqnPN in Γ such that

en “ αnγng´1

fnĂfn . (10.10)

Let α0 “ id and e0 “ γ0 g´1

f0

rf0. Since op rf0q “ g f0ry0 by the construction of the lifts

and since γ0 P Gy0 “ ΓĂy0, we have ope0q “ ry0. Since the stabiliser of g ´1

f0

rf0 is ρ f0pGf0q, the

edge e0 does not depend on the choice of the representative γ0 modulo ρ f0pGf0q on the right,

but depends on the choice of the representative γ0 modulo G1y0“ Γ1

Ăy0on the left.

The hypothesis that if f0 P EY1 then γ0 R G1y0ρ f0pGf0q ensures that the edge e0 does not

belong to EX1. Indeed, assume otherwise that e0 belongs to EX1. Then f0 “ ppe0q P EY1,and by the assumptions on the choice of lifts, the edges g ´1

f0

rf0 and e0 both belong to EX1.Since they are both mapped to f0 by the map X1 Ñ Y1 “ Γ1zX1, they are mapped one to theother by an element of Γ1

Ăy0“ G1y0

. Let γ10 P Γ1Ăy0

be such that γ10 e0 “ g ´1f0

rf0. Then γ10´1γ0

belongs to the stabiliser in Γ of the edge g ´1f0

rf0, which is equal to g ´1f0

ΓĂf0g f0

“ ρ f0pGf0q.

Therefore γ0 P G1y0ρ f0pGf0q, a contradiction.

Assume by induction that en and αn are constructed. Define

αn`1 “ αn γn g´1

fngfn

anden`1 “ αn`1 γn`1 g

´1fn`1

Ćfn`1 ,

so that the induction formula (10.10) at rank n ` 1 is satisfied. By the construction of thelifts, since yn`1 “ tpfnq “ opfn`1, we have

Ćyn`1 “ g ´1fn

tpĂfnq “ g ´1fn`1

opĆfn`1q .

Hence, since γn`1 P Gyn`1 fixes Ćyn`1, using the induction formula (10.10) at rank n for thelast equality,

open`1q “ αn`1 γn`1 g´1

fn`1opĆfn`1q “ αn`1 γn`1 Ćyn`1 “ αn`1 Ćyn`1

“ αn`1 g´1fn

tpĂfnq “ αn γn g´1

fntpĂfnq “ tpenq .

In particular, the sequence penqnPN is an edge path in X.Since the stabiliser of g ´1

fn`1

Ćfn`1 is ρ fn`1pGfn`1q, the edge γn`1 g

´1fn`1

Ćfn`1 does notdepend on the choice of the representative of the right coset γn`1. Let us prove that the length2 edge path pg ´1

fnĂfn, γn`1 g

´1fn`1

Ćfn`1q is geodesic. Otherwise, the two edges of this path

are opposite one to another, hence fn`1 “ fn by using the projection p : X Ñ Y, thereforeg fn`1

“ gfn . Thus γn`1 maps g ´1fn

Ăfn to g ´1fn

Ăfn, hence belongs to ρfnpGfnq “ ρ fn`1pGfn`1q,

a contradiction by the assumptions on the random walk.By construction, the element αn`1 of Γ sends the above length 2 geodesic edge path

pg ´1fn

Ăfn, γn`1 g´1

fn`1

Ćfn`1q to pen, en`1q. This implies on the one hand that the edge path

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pen, en`1q is geodesic, and on the other hand that αn`1 is uniquely defined, since the actionof Γ on X is 2-acylindrical.

In particular, penqnPN is the sequence of edges followed by a (discrete) geodesic ray in X,starting from a point of X1 but not by an edge of X1, that is, an element of B1

`X1. Furthermore,this ray is well defined up to the action of Γ1

Ăy0, hence its image, that we denote by Θpωq, is

well defined in Γz

p

GX. Since ppopenqq “ ppĂynq “ yn for all n P N, the commutativity of thediagram (10.9) is immediate.

For every x P V X1, let B1`X1pxq be the subset of B1

`X1 consisting of the elements w withwp0q “ x. By construction, the above map from the subset of random paths in Ω startingfrom y0 to Γ1

Ăy0zB1`X1p ry0q, which associates to pXnqnPN the Γ1

Ăy0-orbit of the geodesic ray with

consecutive edges penqnPN, is clearly a bijection. This bijection maps the measure P to the

normalised skinning measure σ`Dσ`D

, since by homogeneity, the restriction to B1`X1p ry0q of rσ`X1

to B1`X1p ry0q, normalized to be a probability measure, is the restriction to B1

`X1p ry0q of theAutpXqx-homogeneous probability measure on the space of geodesic rays with origin x in theregular tree X. This proves the result. l

When Y is finite, all the groups Gy for y P V Y are trivial and Y1 is reduced to a vertex,5 theabove random walk is the non-backtracking simple random walk on the nonbipartite regularfinite graph Y, and 1

VolpY,G˚q volY,G˚ is the uniform distribution on V Y. Hence this result(stated as Corollary 1.3 in the Introduction) is classical. See for instance [OW, Thm. 1.2] and[ABLS], which under further assumptions on the spectral properties of Y gives precise ratesof convergence, and also the book [LyP2], including its Section 6.3 and its references.

10.4 Rate of equidistribution for metric and simplicial trees

In this Section, we give error terms for the equidistribution results stated in Theorem 10.2 formetric trees, and in Theorem 10.4 for simplicial trees, under additional assumptions requiredin order to get the error terms for the mixing property discussed in Chapter 9.

We first consider the simplicial case, for the discrete time geodesic flow. Let X, X, Γ, rc,ĂFc, δc, pµ˘x qxPV X, mc “ mFc be as in Section 10.3.

Theorem 10.8. Assume that δc is finite and that the Gibbs measure mc on ΓzGX is finite.Assume furthermore that

(1) the families pΛΓ, µ´x , dxqxPV C ΛΓ and pΛΓ, µ`x , dxqxPV C ΛΓ of metric measure spaces areuniformly doubling,6

(2) there exists α P s0, 1s such that the discrete time geodesic flow on pΓzGX,mcq is expo-nentially mixing for the α-Hölder regularity.

Let D “ pDiqiPI be a locally finite Γ-equivariant family of nonempty proper simplicial subtreesof X with ΓzI finite. Let Ω “ pΩiqiPI be a locally finite Γ-equivariant family of measurablesubsets of

p

GX, with Ωi Ă B1˘Di for all i P I. Assume that σ˘Ω is finite and nonzero. Then

there exists κ1 ą 0 such that for all ψ P C αc pΓz

p

GXq, we have, as nÑ `8,1

pg˘nq˚σ˘Ω

ż

ψ dπ˚pg˘nq˚σ

˘Ω “

1

mc

ż

ψ dmc `Opψα e´κ1 nq .

5The result for general Y1 follows by averaging.6See Section 4.1 for definitions.

152 19/12/2016

Remarks. (1) If rc “ 0, if the simplicial tree X1 with |X1|1 “ C ΛΓ is uniform, if LΓ “ Z andif Γ is a lattice of X1, then we claim that δc “ δΓ is finite, mc “ mBM is finite and mixing,and pΛΓ, µx “ µ˘x , dxqxPV C ΛΓ is uniformly doubling.

Indeed, the above finiteness and mixing properties follow from the results of Section 4.4.Since X1 is uniform, it has a cocompact discrete sugbroup Γ1 whose Patterson density (forthe vanishing potential) is uniformly doubling on ΛΓ1 “ ΛΓ, by Lemma 4.2 (4). Since rc “ 0and Γ is a lattice, the Patterson densities of Γ and of Γ1 coincide (up a scalar multiple) byProposition 4.14 (2).

(2) Assume that rc “ 0, that the simplicial tree X1 with |X1|1 “ CλΓ is uniform withoutvertices of degree 2, that LΓ “ Z and that Γ is a geometrically finite lattice of X1. Then allassumptions of Theorem 10.8 are satisfied by the first remark and by Corollary 9.6. Thereforewe have an exponentially small error term in the equidistribution of the equidistant levelssets.

Proof. We only give the proof when ˘ “ `, the other case is treated similarly. We followthe proof of Theorem 10.2, concentrating on the new features. We now have ΓzI finite byassumption. Let η P s0, 1s and ψ P C α

c pΓz

p

GXq. We consider the constant R ą 0, the testfunction Φη, the fundamental domain ∆Γ and the lift rψ “ ψ ˝ p as in the proof of Theorem10.2.

For all n P N, all w P G`X isometric on r´n,`8r and all ` P V `w, η, e´nR

“ B`pw, e´nRq,7

by Lemma 2.7 where we can take η “ 0, we have dp`, wq “ Ope´nq. Since p is Lipschitz,the map rψ is α-Hölder with α-Hölder norm at most ψα. Hence for all n P N, all w P G`Xisometric on r´n,`8r and all ` P V `

w, η, e´nR, we have

rψp`q “ rψpwq `Ope´nα ψαq . (10.11)

As in the proof of Theorem 10.2 with t replaced by n, using Equation (10.11) instead ofEquation (10.6) in the series of equations (10.7), since the symbols w that appear in them areindeed generalised geodesic lines isometric on r´n,`8r , we have

ş

Γz

p

GX ψ dpgnq˚σ`Ω

pgnq˚σ`Ω

“

ş

ΓzGX ψ Φη ˝ g´n dmc

ş

ΓzGX Φη dmc`Ope´nα ψαq . (10.12)

Let us now apply the assumption on the decay of correlations. In order to do that, weneed to regularise our test functions Φη. We start by an independent lemma.

Lemma 10.9. There are universal constants ε0 ą 0, c0 ě 1 such that for all ε P s0, ε0r and` P GX, the ball Bdp`, εq is contained in

t`1 P GX : `1p0q “ `p0q, `1p˘8q P Bd`p0qp`p˘8q, c0

?ε qu

and contains t`1 P GX : `1p0q “ `p0q, `1p˘8q P Bd`p0qp`p˘8,1c0

?ε qu.

Proof. If `, `1 P GX have distinct footpoints, then dp`p0q, `1p0qq ě 1, so that dp`ptq, `1ptqq ě 14

if |t| ď 14 , so that dp`, `1q ě

ş

14

´ 14

14 e

´2|t| “ ε0.

7We have V `w, η, s “ B`pw, sq for every s ą 0 since X is simplicial and η ă 1.

153 19/12/2016

Conversely, assume that `, `1 P GX have equal footpoints, so that they coincide on r´N,N 1sfor some N,N 1 P N. By the definition of the visual distances (see Equation (2.1)), we have

d`p0qp`p`8q, `1p`8qq “ e´N

1

and similarly d`p0qp`p´8q, `1p´8qq “ e´N . By the definition of the distance on

p

GX (seeEquation (2.4)), we have, by an easy change of variables,

dp`, `1q “

ż `8

N 12 |t´N 1| e´2t dt`

ż ´N

´8

2 | ´N ´ t| e2t dt

“ pe´2N 1 ` e´2N q

ż `8

02u e´2u du “

1

2pe´2N 1 ` e´2N q .

The result follows. l

By the definition of the Gibbs measures,8 this lemma implies that

µ´`p0q`

Bd`p0qp`p´8q,1

c0

?ε q˘

µ``p0q`

Bd`p0qp`p`8q,1

c0

?ε q˘

ď rmcpBdp`, εqq

ď µ´`p0q`

Bd`p0qp`p´8q, c0

?ε q˘

µ``p0q`

Bd`p0qp`p`8q, c0

?ε q˘

.

Since the Patterson densities are uniformly doubling for basepoints in CλΓ, since the foot-points of the geodesic lines in the support of mc belong to CλΓ, the Gibbs measure mc ishence doubling on its support.

As in the proof of [PaP14a, Thm. 20], using discrete convolution approximation (seefor instance [Sem, p. 290-292] or [KinKST]), there exists κ2 ą 0 and, for every η ą 0, anonnegative function RΦη P Cα

c pΓzGXq such that

(1)ş

ΓzGX RΦη dmc “ş

ΓzGX Φη dmc,

(2)ş

ΓzGX | RΦη ´ Φη| dmc “ O`

ηş

ΓzGX Φη dmc

˘

,

(3) RΦηα “ O`

η´κ2 ş

ΓzGX Φη dmc

˘

.

By Equation (10.5), the integralş

ΓzGX Φη dmc “ σ`Ω is constant (in particular indepen-

dent of η). Let mc “mcmc

. All integrals below besides the first one being over ΓzGX, andusing

‚ Equation (10.12) and the above property (2) of the regularised map RΦη for the firstequality,

‚ the assumption of exponential decay of correlations for the second one, involving someconstant κ ą 0, for the second equality,

‚ the above properties (1) and (3) of the regularised map RΦη for the last equality,

8See Equation (4.3)

154 19/12/2016

we hence haveş

Γz

p

GX ψ dpgnq˚σ`Ω

pgnq˚σ`Ω

“

ş

ψ RΦη ˝ g´n dmc

ş

Φη dmc`Ope´nα ψα ` η ψ8q

“

ş

RΦη dmc

ş

ψ dmcş

Φη dmc`Ope´nα ψα ` η ψ8 ` e

´κn RΦηαψαq

“

ż

ψ dmc `O`

pe´nα ` η ` e´κn η´κ2

qψα˘

.

Taking η “ e´λn with λ “ κ2κ2 , the result follows with κ1 “ mintα, κ

2κ2 ,κ2 u. l

Let us now consider the metric tree case, for the continuous time geodesic flow, where themain change is to assume a superpolynomial decay of correlations and hence get a superpoly-nomial error term, for observables which are smooth enough along the flow lines. Let pX, λq,X, Γ, rF , δF pµ˘x qxPX and mF be as in the beginning of Section 4.4.

Theorem 10.10. Assume that δF is finite and that the Gibbs measure mF on ΓzGX is finite.Assume furthermore that

(1) the families pΛΓ, µ´x , dxqxPC ΛΓ and pΛΓ, µ`x , dxqxPC ΛΓ of metric measure spaces are uni-formly doubling,9, and rF is bounded on T 1C ΛΓ,

(2) there exists α P s0, 1s such that the (continuous time) geodesic flow on pΓzGX,mF q hassuperpolynomial decay of α-Hölder correlations.

Let D “ pDiqiPI be a locally finite Γ-equivariant family of nonempty proper closed convexsubsets of X with ΓzI finite. Let Ω “ pΩiqiPI be a locally finite Γ-equivariant family ofmeasurable subsets of

p

GX, with Ωi Ă B1˘Di for all i P I. Assume that σ˘Ω is finite and

nonzero. Then for every n P N, there exists k P N such that for all ψ P C k, αc pΓz

p

GXq, wehave, as tÑ `8,

1

pg˘tq˚σ˘Ω

ż

ψ dπ˚pg˘tq˚σ

˘Ω “

1

mF

ż

ψ dmF `Opψk, α t´nq .

Remarks. (1) If F “ 0, if the metric subtree X 1 “ C ΛΓ of X is uniform, if the lengthspectrum of Γ on X is not contained in a discrete subgroup of R and if Γ is a lattice of X 1, thenwe claim that δF “ δΓ is finite, mF “ mBM is finite and mixing, and pΛΓ, µx “ µ˘x , dxqxPCΛΓ

is uniformly doubling.Indeed, the above finiteness and mixing properties follow from Proposition 4.14 and The-

orem 4.8. Since X 1 is uniform, it has a cocompact discrete sugbroup of isometries Γ1 whosePatterson density (for the vanishing potential) is uniformly doubling on ΛΓ1 “ ΛΓ, by Lemma4.2 (4). Since F “ 0 and Γ is a lattice, the Patterson densities of Γ and of Γ1 coincide (up toa scalar multiple) by Proposition 4.14.

(2) Assume that F “ 0, that the metric subtree X 1 “ C ΛΓ of X is uniform, that thelength spectrum of Γ on X is 4-Diophantine and that Γ is a geometrically finite lattice ofX1. Then all assumptions of Theorem 10.10 are satisfied by the first remark and by Corollary

9See Section 4.1 for definitions.

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9.10. Therefore we have a superpolynomially small error term in the equidistribution of theequidistant levels sets.

Proof. The proof is similar to the one of Theorem 10.8, except for the doubling propertyof the Gibbs measure on its support and the conclusion of the proof. Let X 1 “ C ΛΓ. Themodification of Lemma 10.9 is the third assertion of the following result, which will also beuseful in Section 12.6. Its second claim strengthens for R-trees the Hölder-continuity of thefootpoint projection stated in Section 2.3. If a and b are positive functions of some parameters,we write a — b if there exists a universal constant C ą 0 such that 1

C b ď a ď C b.

Lemma 10.11. Let Y be an R-tree.

(1) There exists a universal constant c1 ą 0 such that for all `, `1 P GY , if dp`, `1q ď c1, then`1p0q P `pRq, the intersection `pRq X `1pRq is not reduced to a point, the orientations of` and `1 coincide on this intersection, and

dp`, `1q — d`p0qp`p´8q, `1p´8qq2 ` d`p0qp`p`8q, `

1p`8qq2 ` dp`p0q, `1p0qq .

(2) The footpoint map π : GY Ñ Y defined by ` ÞÑ `p0q is (uniformly locally) Lipschitz.

(3) There are universal constants ε0 ą 0, c0 ě 1 such that for all ε P s0, ε0r and ` P GY , theball Bdp`, εq in GY is contained in

t`1 P GY : `1p0q P `pRq, dp`1p0q, `p0qq ď c0 ε, `1p˘8q P Bd`p0qp`p˘8q, c0

?ε qu

and contains

t`1 P GY : `1p0q P `pRq, dp`1p0q, `p0qq ď1

c0ε, `1p˘8q P Bd`p0qp`p˘8q,

1

c0

?ε qu .

Proof. (1) Let `, `1 P GY . If `1p0q R `pRq, then `1ptq R `pRq for all t ě 0 or `1ptq R `pRq for allt ď 0, since Y is an R-tree. In the first case, we hence have dp`ptq, `1ptqq ě t for all t ě 0, thusdp`, `1q is at least c2 “

ş`8

0 te´2 t dt “ 14 ą 0. The same estimate holds in the second case.

This argument furthermore shows that if the geodesic segment (or ray or line) `pRq X `1pRq isreduced to a point, then dp`, `1q is at least c2 ą 0.

If dp`1p0q, `p0qq ě 1, then dp`ptq, `1ptqq ě 14 for |t| ď 1

4 , thus

dp`, `1q ě

ż 14

´ 14

1

4e´2|t| dt ,

which is a positive universal constant.If dp`1p0q, `p0qq ď 1, if `pRq X `1pRq contains `1p0q and is not reduced to a point, but if

the orientations of ` and `1 do not coincide on this intersection, then dp`ptq, `1ptqq ě 2t ´dp`p0q, `1p0qq ě 2t ´ 1 for all t ě 1, so that dp`, `1q is at least

ş`8

1 p2t ´ 1q e´2t dt, which is apositive constant.

Assume now that `1p0q P `pRq, that dp`1p0q, `p0qq ď 1, that `pRq X `1pRq is not reducedto a point and that the orientations of ` and `1 coincide on this intersection. Then thereexists s P R such that `1p0q “ `psq, so that |s| “ dp`p0q, `1p0qq. Assume for instance thats ě 0, the other case being treated similarly. Then there exist S, S1 ě 0 maximal such that`1ptq “ `pt` sq for all t P r´S, S1s, with the convention that S “ `8 if `1p´8q “ `p´8q, thatS1 “ `8 if `1p`8q “ `p`8q, and that e´8 “ 0.

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`p0q `pS1 ` sq`psq`p´S ` sq`p´8q `p`8q

`1p´8q `1p`8q

`1p´Sq `1p0q `1pS1q

By the definition of the visual distances (see Equation (2.1)), we have, for t big enough,

d`p0qp`p`8q, `1p`8qq “ e

12rpt´S1q`pt´S1´sqs´t — e´S

1

.

Similarly d`p0qp`p´8q, `1p´8qq — e´S .As can be seen in the above picture, we have

dp`ptq, `1ptqq “

$

&

%

´2t´ 2S ` s if t ď ´Ss if ´ S ď t ď S1 ` s2t´ 2S1 ´ s if t ě S1 ` s .

By the definition of the distance onp

GY (see Equation (2.4)), by easy changes of variables,assuming that at least one of S, S1 is larger than some positive constant for the last line, wehave

dp`, `1q “

ż ´S

´8

p´2t´ 2S ` sq e´2|t| dt`

ż S1`s

´Ss e´2|t| dt`

ż `8

S1`sp2t´ 2S1 ´ sq e´2t dt

“ e´2S`s

ż `8

su e´u du` s

ż S1`s

´Se´2|t| dt` e´2S1´s

ż `8

su e´u du

— e´2S ` e´2S1 ` s .

Assertion (1) of Lemma 10.11 follows.

(2) The second assertion follows immediately from the first one.

(3) The first inclusion in the third assertion follows easily from the first one, and thesecond one follows from the argument of its proof, and the fact that if d`p0qp`p`8q, `1p`8qq,d`p0qp`p`8q, `

1p`8qq and dp`p0q, `1p0qq are at most some small positive constant, then `pRqX`1pRq contains `1p0q and is not reduced to a point. l

If the footpoints of `, `1 P GX 1 are at distance bounded by c0 ε0, then by Proposition 3.10(2), since | rF | is bounded on T 1X 1 by assumption, the quantities |C˘ξ p`p0q, `

1p0qq| for ξ P ΛΓ

are bounded by the constant c10 “ c0 ε0 pmaxT 1X 1 |rF ´ δF |q. By the definition of the Gibbs

measures (see Equation (4.3)), Assertion (3) of the above lemma hence implies that if ε ď ε0then

e´2c10 ε µ´`p0q`

Bd`p0qp`p´8q,1

c0

?ε q˘

µ``p0q`

Bd`p0qp`p`8q,1

c0

?ε q˘

ď rmcpBdp`, εqq

ď e2c10 ε µ´`p0q`

Bd`p0qp`p´8q, c0

?ε q˘

µ``p0q`

Bd`p0qp`p`8q, c0

?ε q˘

.

As in the simplicial case, since the Patterson densities are uniformly doubling for basepointsin X 1, the Gibbs measure mc is hence doubling on its support.

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Fix n P N. As in the end of the proof of Theorem 10.8, using the assumption of superpoly-nomial decay of correlations, involving some degree of regularity k in order to have polynomialdecay in t´Nn where N “ rκ2s` 1, instead of the exponential one, we have for all t ě 1

ş

Γz

p

GX ψ dpgtq˚σ`Ω

pgtq˚σ`Ω

“

ż

ψ dmc `O`

pe´t α ` η ` t´Nn η´κ2

qψk, α˘

.

Taking η “ t´n, by the definition of N , we hence haveş

Γz

p

GX ψ dpgtq˚σ`Ω

pgtq˚σ`Ω

“

ż

ψ dmc `O`

t´n ψk, α˘

.

This proves Theorem 10.10. l

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Chapter 11

Equidistribution of commonperpendicular arcs

In this Chapter, we prove the equidistribution of the initial and terminal vectors of com-mon perpendiculars of convex subsets in the universal covering space level for Riemannianmanifolds and for metric and simplicial trees. The results generalise [PaP16c, Thm. 8].

In Sections 11.1 to 11.3, we consider the continuous time situation where the CAT(´1)-space X is either a proper R-tree without terminal point or a complete Riemannian manifoldwith pinched negative curvature at most ´1, and x0 is any basepoint in X. In Section 11.4,X is the geometric realisation of a simplicial tree X, with the discrete time geodesic flow, andx0 is any vertex of X.

Let Γ be a nonelementary discrete group of isometries of X. Let rF be a continuous Γ-invariant potential on T 1X, which is Hölder-continuous if X is a manifold. Assume thatδ “ δΓ, F˘ is positive and let pµ˘x qxPX be Patterson densities for the pairs pΓ, F˘q, withassociated Gibbs measure mF . Let D´ “ pD´i qiPI´ and D` “ pD`j qjPI` be locally finiteΓ-equivariant families of nonempty proper closed convex subsets of X.

For every pi, jq in I´ˆ I` such that the closures D´i and D`j of D´i and D`j in X XB8Xhave empty intersection, let λi,j “ dpD´i , D

`j q be the length of the common perpendicular

from D´i to D`j , and α´i, j P

p

GX its parametrisation: it is the unique map from R to X suchthat α´i, jptq “ α´i, jp0q P D

´i if t ď 0, α´i, jptq “ α´i, jpλi, jq P D

`j if t ě λi, j , and α´i,j |r0, λi,js “ αi, j

is the shortest geodesic arc starting from a point of D´i and ending at a point of D`j . Let

α`i,j “ gλi,jα´i,j . In particular, we have gλi,j

2 α´i,j “ g´λi,j

2 α`i,j .

We now state our main equidistribution result of common perpendiculars between convexsubsets in the continuous time and upstairs settings. We will give the discrete time versionin Section 11.4, and the downstairs version in Chapter 12.

Theorem 11.1. Let X be a proper R-tree without terminal points or a complete Riemannianmanifold with pinched negative curvature at most ´1. Let Γ be a nonelementary discrete groupof isometries of X and let rF be a bounded Γ-invariant potential on X as above. Assume thatthe critical exponent δ is finite and positive, and that the Gibbs measure mF is finite and

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mixing for the geodesic flow on ΓzGX. Then

limtÑ`8

δ mF e´δ t

ÿ

iPI´„, jPI`„, γPΓ

D´i XD`γj“H, λi, γjďt

eş

αi,γjrF

∆α´i, γjb∆α`

γ´1i, j

“ rσ`D´ b rσ´D`

for the weak-star and narrow convergences of measures on the locally compact spacep

GXˆ

p

GX.

Recall that the narrow topology1 on the set MfpY q of finite measures on a Polish space Y isthe smallest topology such that the map from MfpY q to R defined by µ ÞÑ µpgq is continuousfor every bounded continuous function g : Y Ñ R.

The proof of Theorem 11.1 follows that of [PaP16c, Thm.8], which proves this result whenX is a Riemannian manifold with pinched negative curvature at most ´1 and F “ 0. Thefirst two and a half steps work for both trees and manifolds and are given in Section 11.1.The differences begin in Step 3T. After this, the steps for trees are called 3T and 4T andare given in Section 11.2 and the corresponding steps for manifolds are 3M and 4M , given inSection 11.3.

In the special case of D´ “ pγxqγPΓ and D` “ pγyqγPΓ for some x, y P X, this statementgives the following version with potentials of Roblin’s double equidistribution theorem [Rob2,Theo. 4.1.1] when F “ 0, see [PauPS, Theo. 9.1] for general F when X is a Riemannianmanifold with pinched sectional curvature at most ´1.

Corollary 11.2. Assume that rF is bounded and that the Gibbs measure mF is finite andmixing for the geodesic flow on GX. Then

limtÑ`8

δ mF e´δ t

ÿ

γPΓ : dpx,γyqďt

eşγyx

rF ∆γy b∆γ´1x “ µ`x b µ´y

for the weak-star convergence of measures on the compact space pX YB8Xq ˆ pX YB8Xq. l

Here is a version of Theorem 11.1 without the assumption that the critical exponent ofδ “ δΓ, F is positive.

Theorem 11.3. Assume that rF is bounded and that the Gibbs measure mF is finite andmixing for the geodesic flow on GX. Then for every τ ą 0, we have

limtÑ`8

δ mF

1´ e´τ δe´δ t

ÿ

iPI´„, jPI`„, γPΓ

D´i XD`γj“H, t´τăλi, γjďt

eş

αi,γjrF

∆α´i, γjb∆α`

γ´1i, j

“ rσ`D´ b rσ´D`

for the weak-star convergence of measures on the locally compact spacep

GX ˆ

p

GX.

Proof. The claim follows from Theorem 11.1 by replacing F by F ` κ for κ large enough sothat δΓ, F`κ “ δΓ, F ` κ ą 0, and by using a classical geometric series type of arguments, seefor instance [PauPS, Lem. 9.5] for more details. l

1also called weak topology see for instance [DM, p. 71-III] or [Bil, Part]

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11.1 Part I of the proof of Theorem 11.1: the common part

Step 1: Reduction. By additivity, by the local finiteness of the families D˘, and by thedefinition of rσ˘D¯ “

ř

kPI¯„rσ˘D¯k

, we only have to prove, for all fixed i P I´ and j P I`, that

limtÑ`8

δ mF e´δ t

ÿ

γPΓ : 0ăλi, γjďt

eş

αi,γjrF

∆α´i, γjb∆α`

γ´1i, j

“ rσ`D´ib rσ´

D`j(11.1)

for the weak-star convergence of measures onp

GX ˆ

p

GX.Let Ω´ be a Borel subset of B1

`D´i and let Ω` be a Borel subset of B1

´D`j . To simplify

the notation, let

D´ “ D´i , D` “ D`j , α´γ “ α´i, γj , α`γ “ α`γ´1i, j

, λγ “ λi, γj , rσ˘ “ rσ˘D¯

. (11.2)

Assume that Ω´ and Ω` have positive finite skinning measures and that their boundaries inB1`D

´ and B1´D

` have zero skinning measures (for rσ` and rσ´ respectively). Let

IΩ´,Ω`ptq “ δ mF e´δ t

ÿ

γPΓ : 0ăλγďt

α´γ |s0,λγ sPΩ´|s0,λγ s, α

`γ |s´λγ,0sPΩ

`|s´λγ,0s

e

ş

α´γ

rF. (11.3)

We will prove the stronger statement that, for every such Ω˘, we have

limtÑ`8

IΩ´,Ω`ptq “ rσ`pΩ´q rσ´pΩ`q . (11.4)

Step 2: First upper and lower bounds. Using Lemma 7.5 (1), we may fix R ą 0 suchthat ν˘w pV

¯w, η,Rq ą 0 for all η P s0, 1s and w P B1

¯D˘. Let φ¯η “ φ¯

η,R,Ω˘be the test functions

defined in Equation (10.4).For all t ě 0, let

aηptq “ÿ

γPΓ

ż

`PGXφ´η pg

´t2`q φ`η pgt2γ´1`q drmF p`q . (11.5)

As in [PaP16c], the heart of the proof is to give two pairs of upper and lower bounds, as T ě 0is big enough and η P s0, 1s is small enough, of the (Césaro-type) quantity

iηpT q “

ż T

0eδ t aηptq dt . (11.6)

By passing to the universal cover, the mixing property of the geodesic flow on ΓzGX forthe Gibbs measure mF gives that, for every ε ą 0, there exists Tε “ Tε,η ě 0 such that for allt ě Tε, we have

e´ε

mF

ż

GXφ´η drmF

ż

GXφ`η drmF ď aηptq ď

eε

mF

ż

GXφ´η drmF

ż

GXφ`η drmF .

Hence by Lemma 10.1, for all ε ą 0 and η P s0, 1s, there exists cε “ cε,η ą 0 such that forevery T ě 0, we have

e´εeδ T

δ mF rσ`pΩ´q rσ´pΩ`q ´ cε ď iηpT q ď eε

eδ T

δ mF rσ`pΩ´q rσ´pΩ`q ` cε . (11.7)

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Step 3: Second upper and lower bounds. Let T ě 0 and η P s0, 1s. By Fubini’stheorem for nonnegative measurable maps, the definition2 of the test functions φ˘η and theflow-invariance3 of the fibrations f˘

D¯, we have

iηpT q “ÿ

γPΓ

ż T

0eδ t

ż

GXh´η,R ˝ f

`

D´p`q h`η,R ˝ f

´

D`pγ´1`q

1gt2V `η,RpΩ´qp`q 1g´t2V ´η,RpγΩ`qp`q drmF dt . (11.8)

We start the computations by rewriting the product term involving the functions h˘η,R. Forall γ P Γ and ` P U `

D´XU ´

γD`, define (using Equation (2.11))

w´ “ f`D´p`q P G`, 0X and w` “ f´

γD`p`q “ γf´

D`pγ´1`q P G´, 0X . (11.9)

This notation is ambiguous (w´ depends on `, and w` depends on ` and γ), but it makes thecomputations less heavy. By Equations (10.2) and (3.8), we have, for every t ě 0,

h´η,Rpw´q “ h´η,R ˝ g

´t2pgt2w´q “ eşw´pt2q

w´p0qp rF´δq

h´η, e´t2R

pgt2w´q .

Similarly,

h`η,Rpγ´1w`q “ e

şw`p0q

w`p´t2qp rF´δq

h`η, e´t2R

pg´t2w`q .

Hence,

h´η,R ˝ f`

D´p`q h`η,R ˝ f

´

D`pγ´1`q

“ e´δ t eşw´pt2q

w´p0qrF`

şw`p0q

w`p´t2qrFh´η, e´t2R

pgt2w´qh`η, e´t2R

pg´t2w`q . (11.10)

11.2 Part II of the proof of Theorem 11.1: the metric tree case

In this Section, we assume that X is an R-tree and we will consider the manifold case sepa-rately in Section 11.3.

Step 3T. Consider the product term in Equation (11.8) involving the characteristic functions.By Lemma 2.8 (applied by replacing D` by γD`), there exists t0 ě 2 lnR ` 4 such that forall η P s0, 1s and t ě t0, for all ` P GX, if 1gt2V `η,RpΩ

´qp`q 1g´t2V ´η,RpγΩ`qp`q ‰ 0, then the

following facts hold.

(i) By the convexity of D˘, we have ` P U `

D´XU ´

γD`.

(ii) By the definition4 of w˘, we have w´ P Ω´ and w` P γΩ`. The notation pw´, w`qhere coincides with the notation pw´, w`q in Lemma 2.8.

2See Equation (10.4).3See Equation (2.11).4See Equation (11.9).

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(iii) There exists a common perpendicular αγ from D´ to γD`, whose length λγ satisfies

| λγ ´ t | ď 2η ,

whose origin is α´γ p0q “ w´p0q, whose endpoint is α`γ p0q “ w`p0q, such that the pointsw´p t2q and w

`p´ t2q are at distance at most η from `p0q P αγ .

Hence, by Lemma 3.3 and since rF is bounded,

e´2η F 8 eş

αγrFď e

şw´pt2q

w´p0qrF`

şw`p0q

w`p´t2qrFď e2η F 8 e

ş

αγrF. (11.11)

For all η P s0, 1s, γ P Γ and T ě t0, let

Aη,γpT q “

pt, `q P rt0, T s ˆ GX : ` P V `

η, e´t2Rpgt2Ω´q X V ´

η, e´t2Rpγg´t2Ω`q

(

andjη, γpT q “

ij

pt, `qPAη,γpT q

h´η, e´t2R

pgt2w´q h`η, e´t2R

pg´t2w`q dt drmF p`q .

By the above, since the integral of a function is equal to the integral on any Borel set containingits support, and since the integral of a nonnegative function is nondecreasing in the integrationdomain, there hence exists c4 ą 0 such that for all T ě 0 and η P s0, 1s, we have

iηpT q ě ´ c4 ` e´2ηF 8

ÿ

γPΓ : t0`2ďλγďT´2η

α´γ |r0, λγ sPΩ´|r0, λγ s, α

`γ |r´λγ, 0sPγΩ`|r´λγ, 0s

eş

αγrFjη, γpT q ,

and similarly, for every T 1 ě T (later on, we will take T 1 to be T ` 4η),

iηpT q ď c4 ` e2η F 8

ÿ

γPΓ : t0`2ďλγďT`2η

α´γ |r0, λγ sPΩ´|r0, λγ s, α

`γ |r´λγ, 0sPγΩ`|r´λγ, 0s

eş

αγrFjη, γpT

1q .

Step 4T: Conclusion. Let ε ą 0. Let γ P Γ be such that D´ and γD` do not intersectand the length of their common perpendicular satisfies λγ ě t0 ` 2. Let us prove that if ηis small enough and λγ is large enough (with the enough’s independent of γ), then for everyT ě λγ ` 2η, we have

1´ ε ď jη, γpT q ď 1` ε . (11.12)

This estimate proves the claim (11.4), as follows. For every ε ą 0, if η ą 0 is small enough,we have

iηpT ` 2ηq ě ´c4 ` e´2ηF 8p1´ εq

´ IΩ´,Ω`pT q

δ mF e´δ T´

IΩ´,Ω`pt0 ` 2q

δ mF e´δpt0`2q

¯

and by Equation (11.7)

iηpT ` 2ηq ď cε `eε rσ`pΩ´q rσ´pΩ`q

δ mF e´δpT`2ηq.

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Thus, for η small enough,

rσ`pΩ´q rσ´pΩ`q ěe2ε

1´ εIΩ´,Ω`pT q ` op1q

as T Ñ `8, which gives

lim supTÑ`8

IΩ´,Ω`pT q ď rσ`pΩ´q rσ´pΩ`q.

The similar estimate for the lower limit proves the claim (11.4).To prove the claim (11.12), let η P s0, 1s and T ě λγ ` 2η. To simplify the notation, let

rt “ e´t2R, w´t “ gt2w´ and w`t “ g´t2w` .

By the definition of jη, γ , using the inequalities (10.3) (and the comment following them), wehence have

jη, γpT q “

ij

pt,`qPAη,γpT q

h´η, rtpw´t q h

`η, rtpw

`t q dt drmF p`q

“eOpe´`γ 2q

p2ηq2

ij

pt, `qPAη,γpT q

dt drmF p`q

µW`pw´t qpB`pw´t , rtqq µW´pw`t q

pB´pw`t , rtqq. (11.13)

Let xγ be the midpoint of the common perpendicular αγ . Let us use the Hopf parametri-sation of GX with basepoint xγ , denoting by s its time parameter. When pt, `q P Aη,γpT q, wehave, by Proposition 3.10 where we may take κ2 “ 1,

drmF p`q “ eC´`´

pxγ , `p0qq`C```pxγ , `p0qq

dµ´xγ p`´q dµ`xγ p``q ds

“ eOpηqdµ´xγ p`´q dµ`xγ p``q ds . (11.14)

Let Pγ be the plane domain of pt, sq P R2 such that there exist s˘ P s ´ η, ηr withs¯ “

λγ´t2 ˘ s. It is easy to see that Pγ is a rhombus centred at pλγ , 0q whose area is p2ηq2.

Let ξ˘γ be the point at infinity of any fixed geodesic ray from xγ through α˘γ p0q. If A is asubset of GX, we denote by A˘ the subset t`˘ : ` P Au of B8X.

Lemma 11.4. For every t ě t0, we have

pB˘pw¯t , rtqq¯ “ Bdxγ pξ¯γ , R e

´λγ2 q .

Proof. We prove the statement for the negative endpoints, the proof of the claim for pos-itive endpoints is similar. Let `1 P B˘pw´t , rtq, with `1´ ‰ ξ´γ . Let p P X be such thatr`1p0q, ξ´γ r X r`

1p0q, `1´r “ r`1p0q, ps.

p

ξ´γ w´p0q “ α´γ p0q

`1´

xγ

w´p t2q “ `1p0q

164 19/12/2016

Since t ě t0 ą 2 lnR, we have rt ă 1, hence `1p0q “ w´t p0q “ w´pt2q and p P s`1p0q, ξ´γ r .Since t ě t0 ě 1 ě η, we have p P rxγ , ξ´γ r . Hence

dxγ p`1´, ξ

´γ q “ e´dpp, xγq “ e´dpp,w

´p t2qq´p

λγ2´ t

2q ă rt e

´λγ2` t

2 “ Re´λγ2 .

Conversely, if ξ P Bdxγ pξ¯γ , R e

´λγ2 q with ξ ‰ ξ¯γ , let `1 P GX be such that `1p0q “ w´pt2q

and `1´ “ ξ. Let pw´t be the extension of w´t such that p pw´t q´ “ ξ´γ . Then as above, we havedW´pw´t q

p pw´t , `1q ă rt. l

It follows from this lemma that, for all t ě t0, s˘ P s ´ η, ηr and ` P GX, we haveg¯s

¯

` P B˘pw¯t , rtq if and only if dp`p0q, α˘γ p0qq “ s˘` t2 (or equivalently by the definition of

the time parameter s in Hopf’s parametrisation, s˘` t2 “

λγ2 ˘s), and `˘ P Bdxγ pξ

˘γ , R e

´λγ2 q.

Thus,Aη,γpT q “ Pγ ˆBdxγ pξ

´γ , R e

´λγ2 q ˆBdxγ pξ

`γ , R e

´λγ2 q .

To finish Step 4T and the proof of the theorem for R-trees, note that by the definition ofthe skinning measure (using again the Hopf parametrisation with basepoint xγ), by the aboveLemma 11.4 and by Proposition 3.10 where we may take κ2 “ 1, we have

µW˘pw¯t qpB˘pw¯t , rtqq “ eOpηqµ¯xγ pBdxγ pξ

¯γ , R e

´λγ2 qq . (11.15)

Thus, by the above and by Equations (11.13) and (11.14), (and noting that Opηq ` Opηq “Opηq)

jη, γpT q “eOpe´

λγ2 q

4η2eOpηqp2ηq2

µ´xγ pBdxγ pξ´γ , Re

´λγ2 qqµ`xγ pBdxγ pξ

`γ , Re

´λγ2 qq

µ´xγ pBdxγ pξ´γ , Re

´λγ2 qqµ`xγ pBdxγ pξ

`γ , Re

´λγ2 qq

“ eOpη`e´λγ2 q , (11.16)

which gives the inequalities (11.12). l

The effective control on jη, γpT q given by Equation (11.16) is stronger than what is neededin order to prove Equation (11.12) in Step 4T. We will use it in Section 12.6 in order to obtainerror terms.

11.3 Part III of the proof of Theorem 11.1: the manifold case

The proof of Theorem 11.1 for manifolds is the same as for trees until Equation (11.10). Therest of the proof that we give below is more technical than for trees but the structure of theproof is the same. In this Section, X “ ĂM is a Riemannian manifold, and we identify GXwith T 1

ĂM .

Step 3M. Consider the product term in Equation (11.8) involving the characteristic functions.The quantity 1V `η,RpΩ

´qpg´t2vq 1V ´η,RpΩ

`qpγ´1gt2vq is different from 0 (hence equal to 1) if

and only if

v P gt2V `η,RpΩ

´q X γg´t2V ´η,RpΩ

`q “ V `

η, e´t2Rpgt2Ω´q X V ´

η, e´t2Rpγg´t2Ω`q ,

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see Section 2.5 and in particular Equation (2.15). By [PaP16c, Lem. 7] (applied by replacingD` by γD` and w by v), there exists t0, c0 ą 0 such that for all η P s0, 1s and t ě t0, for allv P T 1

ĂM , if 1V `η,RpΩ´qpg´t2vq 1V ´η,RpΩ

`qpγ´1gt2vq ‰ 0, then the following facts hold:

(i) by the convexity of D˘, we have v P U `

D´XU ´

γD`,

(ii) by the definition of w˘ (see Equation (11.9)), we have w´ P Ω´ and w` P γΩ`. Thenotation pw´, w`q here coincides with the notation pw´, w`q in Lemma 2.8,

(iii) there exists a common perpendicular αγ from D´ to γD`, whose length λγ satisfies

| λγ ´ t | ď 2η ` c0 e´t2 ,

whose origin πpv´γ q is at distance at most c0 e´t2 from πpw´q, whose endpoint πpv`γ q

is at distance at most c0 e´t2 from πpw`q, such that both points πpgt2w´q and

πpg´t2w`q are at distance at most η ` c0 e´t2 from πpvq, which is at distance at

most c0 e´t2 from some point pv of αγ .

In particular, using (iii) and the uniform continuity property of the rF -weighted length (seeProposition 3.10 (3) which introduces a constant c2 P s0, 1s), and since rF is bounded, for allη P s0, 1s, t ě t0 and v P T 1

ĂM for which 1V `η,RpΩ´qpg´t2vq 1V ´η,RpΩ

`qpγ´1gt2vq ‰ 0, we have

e

şπpgt2w´q

πpw´qrF`

şπpw`q

πpg´t2w`qrF“ e

şpv

πpv´γ qrF`

şπpv`γ qpv

rF`Oppη`e´t2qc2 q

“ eş

αγrFeOppη`e´λγ 2qc2 q . (11.17)

For all η P s0, 1s, γ P Γ and T ě t0, define Aη,γpT q as the set of pt, vq P rt0, T sˆT 1ĂM such

that v P V `

η, e´t2Rpgt2Ω´q X V ´

η, e´t2Rpγg´t2Ω`q, and

jη, γpT q “

ij

pt, vqPAη,γpT q

h´η, e´t2R

pgt2w´q h`η, e´t2R

pg´t2w`q dt drmF pvq .

By the above, since the integral of a function is equal to the integral on any Borel set containingits support, and since the integral of a nonnegative function is nondecreasing in the integrationdomain, there hence exists c4 ą 0 such that for all T ě 0 and η P s0, 1s, we have

iηpT q ě ´ c4 `ÿ

γPΓ : t0`2`c0ďλγďT´Opη`e´λγ 2q

v´γ PN´Opη`e´λγ 2q

Ω´, v`γ P γN´Opη`e´λγ 2q

Ω`

eş

αγrFjη, γpT q e

´Oppη`e´λγ 2qc2 q ,

and similarly, for every T 1 ě T ,

iηpT q ď c4 `ÿ

γPΓ : t0`2`c0ďλγďT`Opη`e´λγ 2q

v´γ PNOpη`e´λγ 2q

Ω´, v`γ P γNOpη`e´λγ 2q

Ω`

eş

αγrFjη, γpT

1q eOppη`e´λγ 2qc2 q .

We will take T 1 to be of the form T `Opη` e´λγ2q, for a bigger Op¨q than the one appearingin the index of the above summation.

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Step 4M: Conclusion. Let γ P Γ be such that D´ and γD` have a common perpendicularwith length λγ ě t0 ` 2 ` c0. Let us prove that for all ε ą 0, if η is small enough and λγ islarge enough, then for every T ě λγ`Opη`e´λγ2q (with the enough’s and Op¨q independentof γ), we have

1´ ε ď jη, γpT q ď 1` ε . (11.18)

Note that rσ˘pNεpΩ¯qq and rσ˘pN´εpΩ

¯qq tend to rσ˘pΩ¯q as ε Ñ 0 (since rσ˘pBΩ¯q “ 0 asrequired in Step 1). Using Steps 2, 3M and 4M, this will prove Equation (11.4), hence willcomplete the proof of Theorem 11.1.

We say that pĂM,Γ, rF q has radius-continuous strong stable/unstable ball masses if for everyε ą 0, if r ą 1 is close enough to 1, then for every v P T 1

ĂM , if B´pv, 1q meets the support ofµ`W´pvq

, thenµ`W´pvq

pB´pv, rqq ď eεµ`W´pvq

pB´pv, 1qq

and if B`pv, 1q meets the support of µ´W`pvq

, then

µ´W`pvq

pB`pv, rqq ď eεµ´W`pvq

pB`pv, 1qq .

We say that pĂM,Γ, rF q has radius-Hölder-continuous strong stable/unstable ball masses if thereexists c P s0, 1s and c1 ą 0 such that for every ε P s0, 1s, if B´pv, 1q meets the support ofµ`W´pvq

, then

µ`W´pvq

pB´pv, rqq ď ec1εcµ`

W´pvqpB´pv, 1qq

and if B`pv, 1q meets the support of µ´W`pvq

, then

µ´W`pvq

pB`pv, rqq ď ec1εcµ´

W`pvqpB`pv, 1qq .

Note that when F “ 0 and M has locally symmetric with finite volume, the conditionalmeasures on the strong stable/unstable leaves are homogeneous. Hence pĂM,Γ, rF q has radius-Hölder-continuous strong stable/unstable ball masses.

When the sectional curvature has bounded derivatives and when pĂM,Γ, rF q has Hölderstrong stable/unstable ball masses, we will prove a stronger statement: with a constant c7 ą 0and functions Op¨q independent of γ, for all η P s0, 1s and T ě λγ `Opη ` e´λγ2q, we have

jη, γpT q “´

1`O´e´λγ2

2η

¯¯2eOppη`e´λγ 2qc7 q . (11.19)

This stronger version will be needed for the error term estimate in Section 12.3. In order toobtain Theorem 11.1, only the fact that jη, γpT q tends to 1 as firstly λγ tends to `8, secondlyη tends to 0 is needed. A reader not interested in the error term may skip many technicaldetails below.

Given a, b ą 0 and a point x in a metric space X (with a, b, x depending on parameters),we will denote by Bpx, a eOpbqq any subset Y of X such that there exists a constant c ą 0(independent of the parameters) with

Bpx, a e´c bq Ă Y Ă Bpx, a ec bq .

Let η P s0, 1s and T ě λγ `Opη ` e´λγ2q. To simplify the notation, let

rt “ e´t2R, w´t “ gt2w´ and w`t “ g´t2w` .

167 19/12/2016

By the definition of jη, γ , using the inequalities (10.3), we hence have

jη, γpT q “

ij

pt,vqPAη,γpT q

h´η, rtpw´t q h

`η, rtpw

`t q dt drmF pvq

“eOpe´c2λγ 2q

p2ηq2

ij

pt,vqPAη,γpT q

dt drmF pvq

µ´W`pw´t q

pB`pw´t , rtqq µ`

W´pw`t qpB´pw`t , rtqq

. (11.20)

We start the proof of Equation (11.18) by defining parameters s`, s´, s, v1, v2 associatedwith pt, vq P Aη,γpT q.

v0γ

v2v1

v

gt2w´g´t2w`s

s´ ´s`

x0

W´pg´t2w`qW`pgt2w´qW`pv1qW

´pv2q

W`pv0γqW´pv0

γq

We have pt, vq P Aη,γpT q if and only if there exist s˘ P s ´ η, ηr such that

g¯s¯

v P B˘pg˘t2w¯, e´t2Rq .

The notation s˘ coincides with the one in the proof of Lemma 2.8 (where pD`, wq has beenreplaced by pγD`, vq).

In order to define the parameters s, v1, v2, we use the well known local product structure ofthe unit tangent bundle in negative curvature. If v P T 1M is close enough to v0

γ (in particular,v´ ‰ pv0

γq` and v` ‰ pv0γq´), then let v1 “ f`

HB´pv0γqpvq be the unique element of W´pv0

γq

such that v1` “ v`, let v2 “ f´HB`pv0

γqpvq be the unique element of W`pv0

γq such that v2´ “ v´,and let s be the unique element of R such that g´sv P W`pv1q. The map v ÞÑ ps, v1, v2q isa homeomorphism from a neighbourhood of v0

γ in T 1ĂM to a neighbourhood of p0, v0

γ , v0γq in

RˆW´pv0γq ˆW

`pv0γq. Note that if v “ grv0

γ for some r P R close to 0, then

w´ “ v´γ , w` “ v`γ , s “ r, v1 “ v2 “ v0

γ , s´ “

λγ ´ t

2` s, s` “

λγ ´ t

2´ s .

Up to increasing t0 (which does not change Step 4, up to increasing c4), we may assume thatfor every pt, vq P Aη,γpT q, the vector v belongs to the domain of this local product structureof T 1

ĂM at v0γ .

The vectors v, v1, v2 are close to v0γ if t is large and η small, as the following result shows.

We denote (also) by d the Riemannian distance induced by Sasaki’s metric on T 1ĂM .

Lemma 11.5. For every pt, vq P Aη,γpT q, we have dpv, v0γq, dpv

1, v0γq, dpv

2, v0γq “ Opη` e´t2q.

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Proof. Consider the distance d1 on T 1ĂM , defined by

@ v1, v2 P T1ĂM, d1pv1, v2q “ max

rPr´1,0sd`

πpgrv1q, πpgrv2q

˘

.

As seen in (iii) of Step 3M, we have dpπpw˘q, πpv˘γ qq, dpπpvq, αγq “ Ope´t2q, and furthermore,dpπpgt2w´q, πpvqq,

λγ2 ´

t2 “ Opη ` e´t2q. Hence dpπpvq, πpv0

γqq “ Opη ` e´t2q. By [PauPS,Lem. 2.4], we have

dpπpg´t2´s´vq, πpv´γ qq ď dpπpg´

t2´s´vq, πpw´qq ` dpπpw´q, πpv´γ qq ď R` c0 e

´t2 .

By an exponential pinching argument, we hence have d1pv, v0γq “ Opη ` e´λγ2q. Since d and

d1 are equivalent (see [Bal, page 70]), we therefore have dpv, v0γq “ Opη ` e´λγ2q.

For all w P T 1ĂM and V P TwT

1ĂM , we may uniquely write V “ V ´ ` V 0 ` V ` with

V ´ P TwW´pwq, V 0 P R d

dt |t0gtw and V ` P TwW`pwq. By [PauPS, §7.2] (building on [Brin]

whose compactness assumption on M and torsion free assumption on Γ are not necessary forthis, the pinched negative curvature assumption is sufficient), Sasaki’s metric (with norm ¨ )is equivalent to the Riemannian metric with (product) norm

V 1 “a

V ´ 2 ` V 0 2 ` V ` 2 .

By the dynamical local product structure of T 1ĂM in the neighbourhood of v0

γ and by thedefinition of v1, v2, the result follows, since the exponential map of T 1

ĂM at v0γ is almost

isometric close to 0 and the projection to a factor of a product norm is Lipschitz. l

We now use the local product structure of the Gibbs measure to prove the following result.

Lemma 11.6. For every pt, vq P Aη,γpT q, we have

dt drmF pvq “ eOppη`e´λγ 2qc2 q dt ds dµW´pv0γqpv1q dµW`pv0

γqpv2q .

Proof. By the definition of the measures (see Equations (4.3) and (7.9)), since the aboveparameter s differs, when v´, v` are fixed, only up to a constant from the time parameter inHopf’s parametrisation, we have

drmF pvq “ eC´v´px0, πpvqq`C

`v`px0, πpvqq dµ´x0

pv´q dµ`x0pv`q ds

dµW´pv0γqpv1q “ e

C`v1`

px0, πpv1qqdµ`x0

pv1`q ,

dµW`pv0γqpv2q “ e

C´v2´

px0, πpv2qqdµ´x0

pv2´q .

By Proposition 3.10 (2) since F is bounded, we have |C˘ξ pz, z1q | “ Opdpz, z1qc2q for all ξ P

B8ĂM and z, z1 P ĂM with dpz, z1q bounded. Since the map π : T 1ĂM Ñ ĂM is Lipschitz, and

since v` “ v1` and v´ “ v2´, the result follows from Lemma 11.5 and the cocycle property(3.7). l

When λγ is large, the submanifold gλγ2Ω´ has a second order contact at v0γ with W´pv0

γq

and similarly, g´λγ2Ω` has a second order contact at v0γ with W`pv0

γq. Let Pγ be the planedomain of pt, sq P R2 such that there exist s˘ P s ´ η, ηr with s¯ “

λγ´t2 ˘ s ` Ope´λγ2q.

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Note that its area is p2η `Ope´λγ2qq2. By the above, we have (with the obvious meaning ofa double inclusion)

Aη,γpT q “ Pγ ˆB´pv0

γ , rλγ eOpη`e´λγ 2qq ˆB`pv0

γ , rλγ eOpη`e´λγ 2qq .

By Lemma 11.6, we hence haveż

Aη,γpT qdt drmF pvq “ eOppη`e´λγ 2qc2 q p2η `Ope´λγ2qq2 ˆ

µW´pv0γqpB´pv0

γ , rλγ eOpη`e´λγ 2qqq µW`pv0

γqpB`pv0

γ , rλγ eOpη`e´λγ 2qqq .

(11.21)

The last ingredient of the proof of Step 4M is the following continuity property of strongstable and strong unstable ball volumes as their centre varies. See [Rob2, Lem. 1.16], [PauPS,Prop. 10.16] for related properties, although we need a more precise control for the error termin Section 12.3.

Lemma 11.7. Assume that pĂM,Γ, rF q has radius-continuous strong stable/unstable ball mas-ses. There exists c5 ą 0 such that for every ε ą 0, if η is small enough and λγ large enough,then for every pt, vq P Aη,γpT q, we have

µW´pw`t qpB´pw`t , rtqq “ eOpεc5 q µW´pv0

γqpB´pv0

γ , rλγ qq

andµW`pw´t q

pB`pw´t , rtqq “ eOpεc5 q µW`pv0γqpB`pv0

γ , rλγ qq .

If we furthermore assume that the sectional curvature of ĂM has bounded derivatives andthat pĂM,Γ, rF q has radius-Hölder-continuous strong stable/unstable ball masses, then we mayreplace ε by pη ` e´λγ2qc6 for some constant c6 ą 0.

Proof. We prove the (second) claim for W `, the (first) one for W ´ follows similarly. Thefinal statement is only used for the error estimates in Section 12.3.

v´γ

w´ t2

`γ2v0γ

w´tOpη ` e´`γ2qOpe´`γ2q v´γ

w´

B`pw´, Rq

B`pv´γ , R eOpη`e`γ 2qq

Using respectively Equation (2.13) since w´t “ gt2w´ and rt “ e´t2R, Equation (7.11)where pv, t, wq is replaced by pgt2v, t2, gt2w´q, and Equation (3.8), we have

µW`pw´t qpB`pw´t , rtqq “

ż

vPB`pw´, RqdµW`pgt2w´qpg

t2vq

“

ż

vPB`pw´, RqeC

´v´pπpvq, πpgt2vqq dµW`pw´qpvq

“

ż

vPB`pw´, Rqeşπpgt2vqπpvq

p rF´δqdµW`pw´qpvq . (11.22)

170 19/12/2016

Similarly, for every a ą 0, we have

µW`pv0γqpB`pv0

γ , artqq “

ż

vPB`pv0γ , aRq

eşπpgt2vqπpvq

p rF´δqdµW`pv0

γqpvq . (11.23)

Let h´ : B`pw´, Rq ÑW`pv´γ q be the map such that ph´pvqq´ “ v´, which is well definedand a homeomorphism onto its image if λγ is large enough (since R is fixed). By Proposition7.4 applied with D “ HB`pw

´q and D1 “ HB`pv´γ q, we have, for every v P B`pw´, Rq,

dµW`pw´qpvq “ e´C´v´pπpvq, πph´pvqqq dµW`pv´γ q

ph´pvqq .

Let us fix ε ą 0. The strong stable balls of radius R centred at w´ and v´γ are very close(see the picture in the beginning of the proof). More precisely, recall that R is fixed, andthat dpπpw´q, πpv´γ qq “ Ope´λγ2q and dpπpgt2w´q, πpgλγ2v´γ qq “ Opη ` e´λγ2q. There-fore we have dpπpvq, πph´pvqqq ď ε for every v P B`pw´, Rq if η is small enough and λγ islarge enough. If we furthermore assume that the sectional curvature has bounded deriva-tives, then by Anosov’s arguments, the strong stable foliation is Hölder-continuous, see forinstance [PauPS, Theo. 7.3]. Hence we have dpπpvq, πph´pvqqq “ Oppη ` e´λγ2qc5q for ev-ery v P B`pw´, Rq, for some constant c5 ą 0, under the additional regularity asumption onthe curvature. We also have h´pB`pw´, Rqq “ B`pv´γ , R e

Opεqq and, under the additionalhypothesis on the curvature, h´pB`pw´, Rqq “ B`pv´γ , R e

Oppη`e´λγ 2qc5 qq.In what follows, we assume that ε “ pη ` e´λγ2qc5 under the additional assumption on

the curvature. By Proposition 3.10 (2), we hence have, for every v P B`pw´, Rq,

dµW`pw´qpvq “ eOpεc2 q dµW`pv´γ qph´pvqq

and, using Proposition 3.10 (3),ż πpgt2vq

πpvqp rF ´ δq ´

ż πpgt2h´pvqq

πph´pvqqp rF ´ δq “ Opεc2q .

The result follows, by Equation (11.22) and (11.23) and the continuity property in the radius.l

Now Lemma 11.7 (with ε as in its statement, and when its hypotheses are satisfied) impliesthat

ij

pt,vqPAη,γpT q

dt drmF pvq

µ´W`pw´t q

pB`pw´t , rtqq µ`

W´pw`t qpB´pw`t , rtqq

“eOpεc5 q

ť

pt,vqPAη,γpT qdt drmF pvq

µ´W`pv0

γqpB`pv0

γ , rtqq µ`

W´pv0γqpB´pv0

γ , rtqq.

By Equation (11.20) and Equation (11.21), we hence have

jη, γpT q “ eOppη`e´λγ 2qc2 q eεc5 p2η `Ope´λγ2qq2

p2ηq2

under the technical assumptions of Lemma 11.7. The assumption on radius-continuity ofstrong stable/unstable ball masses can be bypassed using bump functions, as explained in[Rob2, page 81]. l

171 19/12/2016

11.4 Equidistribution of common perpendiculars in simplicialtrees

In this Section, we prove a version of Theorem 11.1 for the discrete time geodesic flow onquotients of simplicial trees (and we leave to the reader the version without the assumptionthat the critical exponent of the system of conductances is positive).

Let X be a locally finite simplicial tree without terminal vertices. Let Γ be a nonelementarydiscrete subgroup of AutpXq. Let rc : EXÑ R be a Γ-invariant system of conductances on X,let rFc be its associated potential (see Section 3.5), and let mc “ mFc . Let D´ “ pD´i qiPI´and D` “ pD`j qjPI` be locally finite Γ-equivariant families of nonempty proper simplicialsubtrees of X.

For every edge path α “ pe1, . . . , enq in X, we set

cpαq “nÿ

i“1

rcpeiq .

Theorem 11.8. Assume that the critical exponent δc of rc is positive and that the Gibbsmeasure mc is finite and mixing for the discrete time geodesic flow on ΓzGX. Then

limtÑ`8

eδc ´ 1

eδcmc e

´δc tÿ

iPI´„, jPI`„, γPΓ

D´i XD`γj“H, λi, γjďt

ecpαi,γjq ∆α´i, γjb∆α`

γ´1i, j

“ rσ`D´ b rσ´D`

for the weak-star convergence of measures on the locally compact spacep

GXˆp

GX.

Proof. The proof is a modification of the continuous time proof for metric trees in Sections11.1 and 11.2. Here, we indicate the changes to adapt the proof to the discrete time. We usethe conventions for the discrete time geodesic flow described in Section 2.7.

Note that for all i P I´, j P I`, γ P Γ, the common perpendicular αi,γj is now an edgepath from D´i to D`γj , and that by Proposition 3.11, we have

ş

αi,γjrFc “ cpαi,γjq.

In the definition of the bump functions in Section 10.1, we assume (as we may) that η ă 1,so that for all η1 P s0, 1r and w P B1

¯D˘ such that w¯ P ΛΓ, we have

V ˘w,η,η1 “ B˘pw, η1q ,

see Equation (2.12) and recall that we are only considering discrete geodesic lines. As `p0q “wp0q for every ` P B˘pw, ηq since η ă 1, and as the time is now discrete, Equations (10.1)and (7.12) give

h˘η, η1pwq “1

µW¯pwqpB˘pw, η1qq

. (11.24)

This is a considerable simplification compared with the inequalities of Equation (10.3).In the whole proof, we restrict to t “ n P N, T “ N P N. In Steps 1 and 2, we define

instead of Equation (11.3)

IΩ´,Ω`pNq “ peδc ´ 1q mc e´δcpN`1q

ÿ

γPΓ : 0ăλγďN

α´γ |s0,λγ sPΩ´|s0,λγ s, α

`γ |s´λγ,0sPΩ

`|s´λγ,0s

e

ş

α´γ

rFc,

172 19/12/2016

and instead of Equation (11.5)

aηpnq “ÿ

γPΓ

ż

`PGXφ´η pg

´tn2u`q φ`η pgrn2sγ´1`q drmcp`q .

Equation (11.6) is replaced by

iηpNq “Nÿ

n“0

eδc n aηpnq ,

so that by a geometric sum argument, the pair of inequalities (11.7) becomes

e´εeδc pN`1q

rσ`pΩ´q rσ´pΩ`q

peδc ´ 1q mc´ cε ď iηpNq ď eε

eδc pN`1qrσ`pΩ´q rσ´pΩ`q

peδc ´ 1q mc` cε .

Step 3 is unchanged up to replacingşT0 by

řNn“0, rF by rFc, δ by δc and t2 by either tn2u

or rn2s, so that Equation (11.10) becomes, since tn2u` rn2s “ n,

h´η,R˝f`

D´p`q h`η,R ˝ f

´

D`pγ´1`q

“ e´δc n eşw´ptn2uq

w´p0qrFc`

şw`p0q

w`p´rn2sqrFch´η, e´tn2uR

pgtn2uw´qh`η, e´rn2sR

pg´rn2sw`q .

The proof then follows similarly as in Section 11.2, with the simplifications in the point (iii)that, taking η ă 12, we have λγ equal to t “ n, and the points w´ptn2 uq, w`p´rn2 sq and `p0qare equal. In particular, Equation (11.11) simplifies as

e

şw´pt t2 uq

w´p0qrFc`

şw`p0q

w`p´r t2 sq

rFc“ e

ş

αγrFc ,

thus avoiding the assumption that Fc (or equivalently c, see Section 3.2) is bounded. The endof Step 3T simplifies as

´ c4 ď iηpNq ´ÿ

γPΓ : t0`2ďλγďN

α´γ |r0, λγ sPΩ´|r0, λγ s, α

`γ |r´λγ, 0sPγΩ`|r´λγ, 0s

eş

αγrFc jη, γpNq ď c4 .

The statement of Step 4T now simplifies as

jη,γpNq “ 1 ,

if η ă 12 , and if γ P Γ is such that D´ and γD` do not intersect and λγ is large enough. We

introduce in its proof the slightly modified notation

r´n “ e´tn2

uR, r`n “ e´rn2

sR, w´n “ gtn2

uw´ and w`n “ g´rn2

sw` .

and we now take as xγ the point at distance tn2 u from its origin on the common perpendicularαγ . Equation (11.13) becomes (using Equation (11.24) instead of Equation (10.3))

jη, γpNq “

ij

pn, `qPAη,γpNq

dn drmcp`q

µW`pw´n qpB`pw´n , r

´n qq µW´pw`n q

pB´pw`n , r`n qq

.

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Since `p0q “ xγ if pn, `q P Aη, γpNq, Equation (11.14) simplifies as

drmcp`q “ dµ´xγ p`´q dµ`xγ p``q ds ,

with ds the counting measure on the Hopf parameter s P Z of `. Replacing Pγ with itsintersection with Z2 reduces it to one point pλγ , 0q, and now s “ s˘ “ 0. Lemma 11.4becomes

pB`pw´n , r´n qq´ “ Bdxγ pξ

´γ , R e

´tλγ2

uq, pB´pw`n , r`n qq` “ Bdxγ pξ

`γ , R e

´rλγ2

sq ,

so thatAη,γpNq “ tpλγ , 0qu ˆBdxγ pξ

´γ , R e

´tλγ2

uq ˆBdxγ pξ`γ , R e

´rλγ2

sq .

Finally, since `p0q “ xγ if pn, `q P Aη, γpNq, Equation (11.15) becomes

µW`pw´n qpB`pw´n , r

´n qq “ µ´xγ pBdxγ pξ

´γ , R e

´tλγ2

uqq,

µW´pw`t qpB˘pw`n , rnqq “ µ`xγ pBdxγ pξ

`γ , R e

´rλγ2

sqq .

The last centred equation in Step 4T now reduces to jη, γpT q “ 1. l

For lattices in regular trees, we get more explicit expressions.

Corollary 11.9. Let X be a pq ` 1q-regular simplicial tree (with q ě 2) and let Γ be a latticeof X such that ΓzX is not bipartite. Assume that the Patterson density is normalised to bea family of probability measures. Let D˘ be nonempty proper simplicial subtrees of X withstabilisers ΓD˘ in Γ, such that D˘ “ pγD˘qγPΓΓD˘

is locally finite. Then

limtÑ`8

q ´ 1

q ` 1VolpΓzzXq q´t

ÿ

pα, β, γqPΓΓD´ˆΓΓD`ˆΓ

0ădpαD´, γβD`qďt

∆α´α, γβb∆α`

γ´1α, β

“ rσ`D´ b rσ´D` ,

for the weak-star convergence of measures on the locally compact spacep

GXˆp

GX.If the measure σ´D` is nonzero and finite, then

limtÑ`8

q ´ 1

q ` 1

VolpΓzzXqσ´D`

q´tÿ

γPΓΓD` , 0ădpD´, γD`qďt

∆α´e, γ“ rσ`D´ ,

for the weak-star convergence of measures on the locally compact spacep

GX.

Proof. In order to prove the first claim, we apply Theorem 11.8 with rc ” 0, so that byPropositions 4.14, 4.15, and 8.1 (3), we have δc “ ln q ą 0, mc “ mBM is finite and mixing,and mBM “

qq`1 VolpΓzzXq.

The second claim follows by restricting to α “ β “ e and integrating on an appropriatefundamental domain (note that Equation 11.4 does not require Ω` to be relatively compact,just to have finite measure for rσ´). l

The mixing assumption in Theorem 11.8 implies that the length spectrum LΓ of Γ isequal to Z.5 The next result considers the other case, when only the square of the geodesicflow is mixing, when appropriately restricted. Note that the smallest nonempty Γ-invariantsimplicial subtree of X is uniform, without vertices of degree 2, for instance in the case whenX is pp` 1, q ` 1q-biregular with p, q ě 2 and Γ is a lattice of X.

5In fact, ergodicity is sufficient to have this.

174 19/12/2016

Theorem 11.10. Assume that the smallest nonempty Γ-invariant simplicial subtree of X isuniform, without vertices of degree 2, and that the length spectrum LΓ of Γ is 2Z. Assumethat the critical exponent δc of rc is positive, that the Gibbs measure mc is finite and that itsrestriction to ΓzGevenX is mixing for the square of the discrete time geodesic flow on ΓzGevenX.Then

limtÑ`8

e2 δc ´ 1

2 e2 δcmc e

´δc tÿ

iPI´„, jPI`„, γPΓ

D´i XD`γj“H, λi, γjďt

ecpαi,γjq ∆α´i, γjb∆α`

γ´1i, j

“ rσ`D´ b rσ´D`

for the weak-star convergence of measures on the locally compact spacep

GXˆp

GX.

Proof. We denote by rσ˘D¯, eventhe restriction of rσ˘D¯ to

p

Geven X, and by rσ˘D¯, oddthe restriction

of rσ˘D¯ top

Godd X “p

GX ´p

Geven X. We denote by VevenX the subset of V X consisting of thevertices at even distance from x0, and by VoddX “ V X´ VevenX its complement. The subsetsVevenX and VoddX are Γ-invariant if LΓ “ 2Z by Equation (4.13).

Let us first prove that

limtÑ`8

e2 δc ´ 1

2 e2 δcmc e

´δc tÿ

iPI´„, jPI`„, γPΓ

πpα´i,γjq, πpα`

γ´1i,jq PVevenX

D´i XD`γj“H, λi, γjďt

ecpαi,γjq ∆α´i, γjb∆α`

γ´1i, j

“ rσ`D´, evenb rσ´D`, even

(11.25)

for the weak-star convergence of measures on the locally compact spacep

Geven Xˆp

Geven X.

The proof of this Equation (11.25) is a modification of the proof of the previous Theorem11.8. We now restrict to t “ 2n P N, T “ 2N P N, and we replace rmc by prmcq|GevenX andpgtqtPZ by pg2tqtPZ. Note that since rmc is invariant under the time 1 of the geodesic flow,which maps ΓzGevenX to ΓzGX´ ΓzGevenX, we have

prmcq|GevenX “1

2rmc . (11.26)

Note that for all i P I´, j P I` and γ P Γ, if πpα´i,γjq and πpα`γ´1i,j

q belong to VevenX,then the distance between D´i and γD`j is even (since for all x, y, z in a simplicial tree, if pis the closest point from x to ry, zs, then dpy, zq “ dpy, xq ` dpx, zq ´ 2 dpx, pq ).

In Steps 1 and 2, we now consider Ω˘ two Borel subsets of B1¯D

˘X

p

Geven X, and we defineinstead of Equation (11.3)

IΩ´,Ω`p2Nq “ pe2 δc ´ 1q

mc

2e´2 δcpN`1q

ÿ

γPΓ : 0ăλγď2N, πpα´γ q, πpα`γ q PVevenX

α´γ |s0,λγ sPΩ´|s0,λγ s, α

`γ |s´λγ,0sPΩ

`|s´λγ,0s

e

ş

α´γ

rFc,

175 19/12/2016

and instead of Equation (11.5)

aηp2nq “ÿ

γPΓ

ż

`PGevenXφ´η pg

´2tn2u`q φ`η pg2rn2sγ´1`q drmcp`q .

Equation (11.6) is replaced by

iηp2Nq “Nÿ

n“0

eδc 2n aηp2nq .

The mixing property of the square of the geodesic flow on ΓzGevenX for the restriction of theGibbs measure mc gives that, for every ε ą 0, there exists Tε “ Tε,η ě 0 such that for alln ě Tε, we have

e´εş

GevenX φ´η drmc

ş

GevenX φ`η drmc

prmcq|GevenX

ď aηp2nq

ďeε

ş

GevenX φ´η drmc

ş

GevenX φ`η drmc

prmcq|GevenX.

Note that GevenX is saturated by the strong stable and strong unstable leaves, since two pointsx, y on a given horosphere of centre ξ P B8X are at even distance one from another (equal to2dpx, pq where rx, ξr X ry, ξr “ rp, ξr ). By the disintegration proposition 7.6, when ` rangesover U ˘

D X GevenX, we have

drmF |U ˘D XGevenXp`q “

ż

ρPB1˘DX

p

Geven Xdν¯ρ p`q drσ

˘Dpρq .

Hence the proof of Lemma 10.1 extends to giveż

GevenXφ¯η drmc “ rσ˘evenpΩ

¯q , (11.27)

where in order to simplify notation rσ˘even “ rσ˘D¯, even

.Therefore, by Equations (11.26) and (11.27), and by a geometric sum argument, the pair

of inequalities (11.7) becomes

2 e´εe2δc pN`1qrσ`evenpΩ

´q rσ´evenpΩ`q

pe2δc ´ 1q mc´ cε

ď iηp2Nq

ď2 eεe2δc pN`1q

rσ`evenpΩ´q rσ´evenpΩ

`q

pe2δc ´ 1q mc` cε .

Up to replacing the summations from n “ 0 to N to summations on even numbers between0 to 2N , and replacing tn2u by 2tn2u as well as rn2s by 2tn2u, the rest of the proof appliesand gives the result, noting that in claim (iii) of Step 3T, we furthermore have that the originand endpoint of the constructed common perpendicular αγ are in VevenX. This concludes theproof of Equation (11.25).

176 19/12/2016

The remainder of the proof of Theorem 11.10 consists in proving versions of the equidis-tribution result Equation (11.25) respectively in

p

Goddˆ

p

Godd,p

Gevenˆ

p

Godd,p

Goddˆ

p

Geven, andin summing these four contributions.

By applying Equation (11.25) by replacing x0 by a vertex x10 in VoddX, which exchangesVevenX and VoddX,

p

Geven andp

Godd, as well as rσ˘D¯, evenand rσ˘D¯, odd

, we have

limtÑ`8

e2 δc ´ 1

2 e2 δcmc e

´δc tÿ

iPI´„, jPI`„, γPΓ

πpα´i,γjq, πpα`

γ´1i,jq PVoddX

D´i XD`γj“H, λi, γjďt

ecpαi,γjq ∆α´i, γjb∆α`

γ´1i, j

“ rσ`D´, oddb rσ´D`, odd

(11.28)

for the weak-star convergence of measures on the locally compact spacep

Godd Xˆp

Godd X.

Let us now apply Equation (11.25) by replacing D´ “ pD´i qiPI´ by N1D´ “ pN1D´i qiPI´ .

Let us consider the map ϕ` :

p

GX Ñ

p

GX, which maps a generalised geodesic line ` to thegeneralised geodesic line which coincides with g`1` on r0,`8r and is constant (with value`p1q) on s ´ 8, 0r. Note that this map is continuous and Γ-equivariant, and that it mapsp

Geven X inp

Godd X andp

Godd X inp

Geven X.Furthermore, by convexity, ϕ` induces for every i P I´ an homeomorphism from B1

`D´i

to B1`N1D

´i , which sends B1

`D´i X

p

Godd X to B1`N1D

´i X

p

Geven X, such that, by Equation (7.8),for all w P B1

`D´i X

p

Godd X, if ew is the first (respectively the last) edge followed by w

drσ`D´i , odd

pwq “ ecpewq´δc drσ`N1D

´i , even

pϕ`pwqq .

Note that for all ` ą 0, there is a one-to-one correspondance between the set of commonperpendiculars of length `, with origin and endpoint both in Veven, between N1D

´i and γD`j

for all i P I´, j P I` and γ P Γ, and the set of common perpendiculars of length `` 1, withorigin in Vodd and endpoint in Veven, between D´i and γD`j for all i P I´, j P I` and γ P Γ.In particular, ϕ`pα´i,γjq is the common perpendicular between N1D

´i and γD`j , starting at

time t “ 0 from N1D´i .

Therefore Equation (11.25) applied by replacing D´ “ pD´i qiPI´ by N1D´ “ pN1D´i qiPI´

gives

limtÑ`8

e2 δc ´ 1

2 e2 δcmc e

´δc t

ÿ

iPI´„, jPI`„, γPΓ

πpα´i,γjq PVoddX, πpα`γ´1i,jq PVevenX

D´i XD`γj“H, λi, γjďt`1

ecpe

α´i, γj

q`cpϕ`pα´i,γjqq

∆α´i, γjb∆α`

γ´1i, j

“ eδc rσ`D´, oddb rσ´D`, even

for the weak-star convergence of measures on the locally compact spacep

Godd X ˆp

Geven X.

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Since cpeα´i, γj q ` cpϕ`pα´i,γjqq “ cpα´i,γjq, replacing t by t´ 1 and simplifying by eδc , we get

limtÑ`8

e2 δc ´ 1

2 e2 δcmc e

´δc t

ÿ

iPI´„, jPI`„, γPΓ

πpα´i,γjq PVoddX, πpα`γ´1i,jq PVevenX

D´i XD`γj“H, λi, γjďt

ecpα´i,γjq ∆α´i, γj

b∆α`γ´1i, j

“ rσ`D´, oddb rσ´D`, even

(11.29)

for the weak-star convergence of measures on the locally compact spacep

Godd Xˆp

Geven X.

Now Theorem 11.10 follows bysumming Equation (11.25), Equation (11.28), Equation (11.29) and the formula, proven

similarly, obtained from Equation (11.25) by replacing D` “ pD`j qjPI` by pN1D`j qjPI` . l

The following result for bipartite graphs (of groups) is used in the arithmetic applicationsin Part III (see Section 15.4).

Corollary 11.11. Let X be a pp`1, q`1q-biregular simplicial tree (with p, q ě 2, possibly withp “ q), with corresponding partition V X “ VpX\ VqX. Let Γ be a lattice of X such that thispartition is Γ-invariant. Assume that the Patterson density is normalised so that µx “ p`1

?p

for every x P VpX. Let D˘ be nonempty proper simplicial subtrees of X with stabilisers ΓD˘in Γ, such that the families D˘ “ pγD˘qγPΓΓD˘

are locally finite. Then

limtÑ`8

pq ´ 1

2TVolpΓzzXq

?pq´t´2

ÿ

pα, β, γqPΓΓD´ˆΓΓD`ˆΓ

0ădpαD´, γβD`qďt

∆α´α, γβb∆α`

γ´1α, β

“ rσ`D´ b rσ´D`

for the weak-star convergence of measures on the locally compact spacep

GXˆp

GX.If the measure σ´D` is nonzero and finite, then

limtÑ`8

pq ´ 1

2

TVolpΓzzXqσ´D`

?pq´t´2

ÿ

γPΓΓD`0ădpD´, γD`qďt

∆α´e, γ“ rσ`D´ ,

for the weak-star convergence of measures on the locally compact spacep

GX.

Proof. In order to prove the first result, we apply Theorem 11.10 with rc ” 0, so that byPropositions 4.14, 4.15, and 8.1 (2), we have δc “ 1

2 lnppqq ą 0, mc “ mBM is finite andits restriction to ΓzGevenX is mixing under the square of the geodesic flow, and mBM “

TVolpΓzzXq.The second claim follows as in the proof of Corollary 11.9. l

Remark. In some special occasions, the measures involved in the statements of Theorem11.10 and Corollary 11.11 (whether skinning measures or Dirac masses) are actually all sup-ported on

p

Geven X (up to choosing appropriately x0). This is in particular the case if X “ rX|1is a simplicial tree and if D˘ “ pγD˘qγPΓΓD˘

with D´,D` at even signed distance (see below),as the following proposition shows.

178 19/12/2016

` ´H H 1

H H 1

The signed distance between horoballs H and H 1 in an R-tree that are not centred at thesame point at infinity is the distance between them (that is, the length of their common per-pendicular) if they are disjoint, or the opposite of the diameter of their intersection otherwise.Note that if nonempty, the intersection of H and H 1 is a ball centred at the midpoint ofthe segment contained in the geodesic line between the two points at infinity of the horoballs,which lies in both horoballs.

Lemma 11.12. Let X be a simplicial tree, Γ a subgroup of AutpXq and H ,H 1 two horoballsin X (whose boundaries are contained in V X), which either are equal or have distinct points atinfinity. If ΛΓ Ă 2Z and H ,H 1 are at even signed distance, then the signed distance betweenH and γH 1 is even for every γ P Γ such that H and γH 1 do not have the same point atinfinity.

Proof. For every horoball H 2 and for all s P N, let H 2rss be the horoball contained inH , whose boundary is at distance s from the boundary of H . Shrinking the horoballs Hand H 1, by replacing them by the horoballs H rss and H 1rss for any s P N, only changes by˘2s the considered signed distances. Hence, taking s big enough, we may assume that Hand γH 1 are disjoint, and that H and H 1 are disjoint or equal. Let rx, x1s be the commonperpendicular between H and H 1 with x P BH , x1 P BH 1, and let ry, y1s be the one betweenH and γH 1, with y P BH , y1 P BpγH 1q. Note that γx1 P BpγH 1q.

x

H

y1y

γx1

γH 1

The distance between two points x, y of a horosphere is always even (equal to twice thedistance from x to the geodesic ray from y to the point at infinity of the horosphere). Sincegeodesic triangles in trees are tripod, for all a, b, c in a simplicial tree, since

dpa, cq “ dpa, bq ` dpb, cq ´ 2dpb, ra, csq ,

if dpa, bq and dpb, cq are even, so is dpa, cq.Since ΛΓ Ă 2Z, the distance between x1 and γx1 is even by Equation (4.13). Since dpx, x1q

is even by assumption, we hence have that dpx, γx1q is even. Therefore

dpy, y1q “ dpx, γx1q ´ dpx, yq ´ dpy1, γx1q

is even. l

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180 19/12/2016

Chapter 12

Equidistribution and counting ofcommon perpendiculars in quotientspaces

In this Chapter, we use the results of Chapter 11 to prove equidistribution and countingresults in Riemannian manifolds (or good orbifolds) and in metric and simplicial graphs (ofgroups).

Let X, x0, Γ and rF be as in the beginning of Chapter 11.

12.1 Multiplicities and counting functions in Riemannian orb-ifolds

In this Section, we assume that X “ ĂM is a Riemannian manifold. We denote its quotientRiemannian orbifold under Γ by M “ ΓzĂM , and the quotient Riemannian orbifold under Γ

of its unit tangent bundle by T 1M “ ΓzT 1ĂM . We use the identifications GX “ G˘, 0X “

T 1X “ T 1ĂM explained in Chapter 2.

Let D “ pDiqiPI be a locally finite Γ-equivariant family of nonempty proper closed convexsubsets of ĂM . Let Ω “ pΩiqiPI be a Γ-equivariant family of subsets of T 1

ĂM , where Ωi is ameasurable subset of B1

˘Di for all i P I (the sign ˘ being constant). The multiplicity of anelement v P T 1M with respect to Ω is

mΩpvq “Card ti P I„ : rv P Ωiu

CardpStabΓ rvq,

for any preimage rv of v in T 1ĂM . The numerator and the denominator are finite by the local

finiteness of the family D and the discreteness of Γ, and they depend only on the orbit of rvunder Γ.

The numerator takes into account the multiplicities of the images of the elements of Ωin T 1M . The denominator of this multiplicity is also natural, as any counting problem ofobjects possibly having symmetries, the appropriate counting function consists in taking asthe multiplicity of an object the inverse of the cardinality of its symmetry group.

Examples 12.1. The following examples illustrate the behaviour of the multiplicity when Γis torsion-free and Ω “ B1

˘D .

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(1) If for every i P I, the quotient ΓDizDi of Di by its stabiliser ΓDi maps injectively in Mby the map induced by the inclusion of Di in ĂM , and if for every i, j P I such that j R Γi, theintersection Di XDj is empty, then the nonzero multiplicities mΩp`q are all equal to 1.

(2) Here is a simple example of a multiplicity different from 0 or 1.Let c be a closed geodesic in the Riemannian manifold M , let rc bea geodesic line in ĂM mapping to c in M , let D “ pγ rcqγPΓ, let x bea double point of c, let v P T 1

xM be orthogonal to the two tangentlines to c at x (this requires the dimension of ĂM to be at least 3, ifx is a transverse self-intersection point). Then mB1

˘Dpvq “ 2.

xv

c

Given t ą 0 and two unit tangent vectors v, w P T 1M , we define the number ntpv, wqof locally geodesic paths having v and w as initial and terminal tangent vectors respectively,weighted by the potential F , with length at most t, by

ntpv, wq “ÿ

α

CardpΓαq eş

α F ,

where the sum ranges over the locally geodesic paths α : r0, ss Ñ M in the Riemannianorbifold M such that 9αp0q “ v, 9αpsq “ w and s P s0, ts, and Γα is the stabiliser in Γ of anygeodesic path rα in ĂM mapping to α by the quotient map ĂM ÑM . If F “ 0 and Γ is torsionfree, then ntpv, wq is precisely the number of locally geodesic paths having v and w as initialand terminal tangent vectors respectively, with length at most t.

Let Ω´ “ pΩ´i qiPI´ and Ω` “ pΩ`j qjPI` be Γ-equivariant families of subsets of T 1ĂM , where

Ω¯k is a measurable subset of B1˘D

¯k for all k P I¯. We will denote by NΩ´,Ω`, F : s0,`8r Ñ R

the following counting function: for every t ą 0, let NΩ´,Ω`, F ptq be the number of commonperpendiculars whose initial vectors belong to the images in T 1M of the elements of Ω´ andterminal vectors to the images in T 1M of the elements of Ω`, counted with multiplicities andweighted by the potential F , that is:

NΩ´,Ω`, F ptq “ÿ

v, wPT 1M

mΩ´pvq mΩ`pwq ntpv, wq .

When Ω˘ “ B1¯D˘, we denote NΩ´,Ω`, F by ND´,D`, F .

Remark 12.2. Let Y be a negatively curved complete connected Riemannian manifold andlet rY Ñ Y be its Riemannian universal cover. Let D˘ be a locally convex1 geodesic metricspace endowed with a continuous map f˘ : D˘ Ñ Y such that if rD˘ Ñ D˘ is a locallyisometric universal cover and if rf˘ : rD˘ Ñ rY is a lift of f˘, then rf˘ is on each connectedcomponent of rD˘ an isometric embedding whose image is a proper nonempty closed locallyconvex subset of rY , and the family of images under the covering group of rY Ñ Y of theimages by rf˘ of the connected components of rD˘ is locally finite. Then D˘ (or the pairpD˘, f˘q) is a proper nonempty properly immersed closed locally convex subset of Y .

If Γ is a discrete subgroup without torsion of isometries of a CAT(´1) Riemannian mani-foldX, if D˘ “ pγ rD˘qγPΓ where rD˘ is a nonempty proper closed convex subset ofX such thatthe family D˘ is locally finite, and if D˘ is the image of rD˘ by the covering map X Ñ ΓzX,then D˘ is a proper nonempty properly immersed closed convex subset of ΓzX. Under theseassumptions, ND´,D`, F is the counting function ND´, D`, F given in the introduction.

1not necessarily connected

182 19/12/2016

Let us continue fixing the notation used in Sections 12.2 and 12.3. For every pi, jq inI´ˆI` such that D´i and D`j have a common perpendicular2, we denote by αi, j this commonperpendicular, by λi, j its length, by v´i, j P B

1`D

´i its initial tangent vector and by v`i, j P B

1´D

`i

its terminal tangent vector. Note that if i1 „ i, j1 „ j and γ P Γ, then

γ αi1, j1 “ αγi, γj , λi1, j1 “ λγi, γj and γ v˘i1, j1 “ v˘γi, γj . (12.1)

When Γ has no torsion, we have, for the diagonal action of Γ on I´ ˆ I`,

ND´,D`, F ptq “ÿ

pi, jqPΓzppI´„qˆpI`„qq : D´i XD`j “H, λi, jďt

eş

αi, jrF.

When the potential F is zero and Γ acts without torsion, ND´,D`, F ptq is the number ofcommon perpendiculars of length at most t, and the counting function t ÞÑ ND´,D`, 0ptq hasbeen studied in various special cases of negatively curved manifolds since the 1950’s and in anumber of recent works, see the Introduction. The asymptotics of ND´,D`, 0ptq as t Ñ `8

in the case when X is a Riemannian manifold with pinched negative curvature are describedin general in [PaP16c, Thm. 1], where it is shown that if the skinning measures σ`D´ and σ´D`are finite and nonzero, then as sÑ `8,

ND´,D`, 0psq „σ`D´ σ

´

D`

mBM

eδΓ s

δΓ. (12.2)

12.2 Common perpendiculars in Riemannian orbifolds

Corollary 12.3 below is the main result of this text on the counting with weights of commonperpendiculars and on the equidistribution of their initial and terminal tangent vectors in neg-atively curved Riemannian manifolds endowed with a Hölder potential. We use the notationof Section 12.1.

The following observation on the behaviour of induced3 measures under quotients byproperly discontinuous group actions will be used in the proof of the following result and alsothose of its analogues in Section 12.4. Let G be a discrete group that acts properly on a Polishspace rY and let Y “ GzrY . Let rµk for k P N and rµ be G-invariant locally finite measures onrY , with finite induced measures µk for k P N and µ on Y . If for every Borel subset B of rYwith rµpBq finite and rµpBBq “ 0 we have limkÑ8 rµkpBq “ rµpBq, then the sequence pµkqkPNnarrowly converges to µ.

Corollary 12.3. Let ĂM be a complete simply connected Riemannian manifold with pinchednegative sectional curvature at most ´1. Let Γ be a nonelementary discrete group of isometriesof ĂM . Let rF : T 1

ĂM Ñ R be a bounded Γ-invariant Hölder-continuous function with positivecritical exponent δ. Let D´ “ pD´i qiPI´ and D` “ pD`j qjPI` be locally finite Γ-equivariantfamilies of nonempty proper closed convex subsets of ĂM . Assume that the Gibbs measure mF

is finite and mixing for the geodesic flow on T 1M . Then,

limtÑ`8

δ mF e´δ t

ÿ

v, wPT 1M

mB1`D´pvq mB1

´D`pwq ntpv, wq ∆v b∆w “ σ`D´ b σ´

D` (12.3)

2that is, whose closures D´i and D`j in X Y B8X have empty intersection3see for instance [PauPS, §2.6] for a definition

183 19/12/2016

for the weak-star convergence of measures on the locally compact space T 1M ˆ T 1M . If σ`D´and σ´D` are finite, the result also holds for the narrow convergence.

Furthermore, for all Γ-equivariant families Ω˘ “ pΩ˘k qkPI˘ of subsets of T 1ĂM with Ω¯k a

Borel subset of B1˘D

¯k for all k P I¯, with nonzero finite skinning measure and with boundary

in B1˘D

¯k of zero skinning measure, we have, as tÑ `8,

NΩ´,Ω`, F ptq „σ`

Ω´ σ´

Ω`

δ mF eδ t .

Proof. Note that the sum in Equation (12.3) is locally finite, hence it defines a locally finitemeasure on T 1M ˆ T 1M . We are going to rewrite the sum in the statement of Theorem 11.1in a way which makes it easier to push it down from T 1

ĂM ˆ T 1ĂM to T 1M ˆ T 1M .

For every rv P T 1ĂM , let

m¯prvq “ Card tk P I¯„ : rv P B1˘D

¯k u ,

so that for every v P T 1M , the multiplicity of v with respect to the family B1˘D¯ is4

mB1˘D¯pvq “

m¯prvq

CardpStabΓ rvq,

for any preimage rv of v in T 1ĂM .

For all γ P Γ and rv, rw P T 1ĂM , there exists pi, jq P pI´„q ˆ pI`„q such that rv “ v´i,γj

and rw “ v`γ´1i,j

“ γ´1v`i,γj if and only if γ rw P gR rv, there exists i1 P I´„ such that rv P B1`D

´i1

and there exists j1 P I`„ such that γ rw P B1´D

`j1 . Then the choice of such elements pi, jq, as

well as i1 and j1, is free. We hence have

ÿ

iPI´„, jPI`„, γPΓ

0ăλi, γjďt , v´i, γj“rv , v

`

γ´1i, j“ rw

eş

αi, γjrF

∆v´i, γjb∆v`

γ´1i, j

“ÿ

γPΓ, 0ăsďtγ rw“gsrv

eşγπp rwqπprvq

rFCard

pi, jq P pI´„q ˆ pI`„q : v´i, γj “ rv , v`

γ´1i, j“ rw

(

∆rv b∆

rw

“ÿ

γPΓ, 0ăsďtγ rw“gsrv

eşγπp rwqπprvq

rFm´prvq m`pγ rwq ∆

rv b∆rw .

Therefore

ÿ

iPI´„, jPI`„, γPΓ0ăλi, γjďt

eş

αi, γjrF

∆v´i, γjb∆v`

γ´1i, j

“ÿ

rv, rw PT 1ĂM

´

ÿ

γPΓ, 0ăsďtγ rw“gsrv

eşγπp rwqπprvq

rF¯

m´prvq m`p rwq ∆rv b∆ rw .

4See Section 12.1

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By definition, σ˘D¯ is the measure on T 1M induced by the Γ-invariant measure rσ˘D¯ . ThusCorollary 12.3 follows from Theorem 11.1 and Equation (11.4) after a similar reduction5 as inSection 11.1, and since no compactness assumptions were made on Ω˘ to get this equation,by [PauPS, §2.6]. l

In particular, if the skinning measures σ`D´ and σ´D` are positive and finite, Corollary 12.3gives, as tÑ `8,

ND´,D`, F ptq „σ`D´ σ

´

D`

δ mF eδ t .

Remark 12.4. Under the assumptions of Corollary 12.3 with the exception that δ may nowbe nonpositive, we have the following asymptotic result as t Ñ `8 for the growth of theweighted number of common perpendiculars with lengths in st´ τ, ts for every fixed τ ą 0:

ND´,D`, F ptq ´ND´,D`, F pt´ τq „p1´ e´δ τ q σ`D´ σ

´

D`

δ mF eδ t .

This result follows by considering a large enough constant σ such that δΓ, F`σ “ δ ` σ ą 0,by applying Corollary 12.3 with the potential F ` σ (see Remark 7.1 (2)), and by an easysubdivision and geometric series argument, see [PauPS, Ch. 9].

Using the continuity of the pushforwards of measures for the weak-star and the narrowtopologies, applied to the basepoint maps π ˆ π from T 1

ĂM ˆ T 1ĂM to ĂM ˆ ĂM , and from

T 1M ˆ T 1M to M ˆM , we have the following result of equidistribution of the ordered pairsof endpoints of common perpendiculars between two equivariant families of convex sets inĂM or two families of locally convex sets in M . When M has constant curvature and finitevolume, F “ 0 and D´ is the Γ-orbit of a point and D` is the Γ-orbit of a totally geodesiccocompact submanifold, this result is due to Herrmann [Her]. When D˘ are Γ-orbits of pointsand F is a Hölder potential, see [PauPS, Thm. 9.1,9.3], and we refer for instance to [BoyM]for an application of this particular case.

Corollary 12.5. Let ĂM,Γ, rF ,D´,D` be as in Corollary 12.3. Then

limtÑ`8

δ mF e´δ t

ÿ

iPI´„, jPI`„, γPΓ0ăλi, γjďt

eş

αi, γjrF

∆πpv´i, γjqb∆πpv`

γ´1i, jq“ π˚rσ

`

D´ b π˚rσ´

D` ,

for the weak-star convergence of measures on the locally compact space ĂM ˆ ĂM , and

limtÑ`8

δ mF e´δ t

ÿ

v, wPT 1M

mB1`D´pvq mB1

´D`pwq ntpv, wq ∆πpvq b∆πpwq

“ π˚σ`

D´ b π˚σ´

D` ,

for the weak-star convergence of measures on M ˆM . If the measures σ˘D¯ are finite, thenthe above claim holds for the narrow convergence of measures on M ˆM . l

5See Step 1 of the proof of Theorem 11.1.

185 19/12/2016

We will now prove Theorems 1.4 and 1.5 (1) in the Introduction for Riemannian manifolds.Recall from Remark 12.2 the definition of proper nonempty properly immersed closed locallyconvex subsets D˘ in a pinched negatively curved complete connected Riemannian manifoldY and the associated maps rf˘ : rD˘ Ñ rY .

Proof of Theorems 1.4 and 1.5 (1) for Riemannian manifolds. Let Y, F,D˘ be as inthese statements and assume that Y is a Riemannian manifold. Let Γ be the covering groupof the universal Riemannian cover rY Ñ Y . Let I˘ “ Γˆπ0p rD

˘q with the action of Γ definedby γ ¨ pα, cq “ pγα, cq for all γ, α P Γ and every component c of rD˘. Consider the familiesD˘ “ pD˘k qkPI˘ where D˘k “ α rf˘pcq if k “ pα, cq. Then D˘ are Γ-equivariant families ofnonempty proper closed convex subsets of rY , which are locally finite since D˘ are properlyimmersed in Y . The conclusions in Theorems 1.4 and 1.5 (1) when Y is a manifold thenfollow from Corollary 12.3, applied with ĂM “ rY and with rF the lift of F to T 1

ĂM . l

Corollary 12.6. Let ĂM,Γ, rF ,D´,D` be as in Corollary 12.3. Assume that σ˘D¯ are finiteand nonzero. Then

limsÑ`8

limtÑ`8

δ mF 2 e´δt

σ`D´σ´

D`

ÿ

vPT 1M

mB1`D´pvq nt,D`pvq ∆gsv “ mF ,

wherent,D`pvq “

ÿ

wPT 1M

mB1´D`pwq ntpv, wq

is the number (counted with multiplicities) of locally geodesic paths in M of length at most t,with initial vector v, arriving perpendicularly to D`.

Proof. For every s P R, by Corollary 12.3, using the continuity of the pushforwards ofmeasures by the first projection pv, wq ÞÑ v from T 1M ˆ T 1M to T 1M , and by the geodesicflow on T 1M at time s, since pgsq˚∆v “ ∆gsv, we have

limtÑ`8

δ mF e´δt

ÿ

vPT 1M

mB1`D´pvq nt,D`pvq ∆gsv “ pgsq˚σ

`

D´σ´

D` .

The result then follows from Theorem 10.2 with Ω “ B1`D´. l

12.3 Error terms for equidistribution and counting for Rieman-nian orbifolds

In Section 9.1, we discussed various results on the rate of mixing of the geodesic flow forRiemannian manifolds. In this Section, we apply these results to give error bounds to thestatements of equidistribution and counting of common perpendicular arcs given in Section12.2. We use again the notation of Section 12.1.

Theorem 12.7. Let ĂM be a complete simply connected Riemannian manifold with pinchednegative sectional curvature at most ´1. Let Γ be a nonelementary discrete group of isome-tries of ĂM . Let rF : T 1

ĂM Ñ R be a bounded Γ-invariant Hölder-continuous function withpositive critical exponent δ. Assume that pĂM,Γ, rF q has radius-Hölder-continuous strong sta-ble/unstable ball masses. Let D´ “ pD´i qiPI´ and D` “ pD`j qjPI` be locally finite Γ-equivariant families of nonempty proper closed convex subsets of ĂM , with finite nonzero skin-ning measure σD´ and σD`. Let M “ ΓzĂM and let F : T 1M Ñ R be the potential induced byrF .

186 19/12/2016

(1) Assume that M is compact and that the geodesic flow on T 1M is mixing with exponentialspeed for the Hölder regularity for the potential F . Then there exist α P s0, 1s and κ1 ą 0 suchthat for all nonnegative ψ˘ P C α

c pT1Mq, we have, as tÑ `8,

δ mF

eδ t

ÿ

v, wPT 1M

mB1`D´pvq mB1

´D`pwq ntpv, wq ψ´pvqψ`pwq

“

ż

T 1Mψ´dσ`D´

ż

T 1Mψ`dσ´D` `Ope´κ

1tψ´α ψ`αq .

(2) Assume that ĂM is a symmetric space, that D˘k has smooth boundary for every k P I˘,that mF is finite and smooth, and that the geodesic flow on T 1M is mixing with exponentialspeed for the Sobolev regularity for the potential F . Then there exist ` P N and κ1 ą 0 suchthat for all nonnegative maps ψ˘ P C `

c pT1Mq, we have, as tÑ `8,

δ mF

eδ t

ÿ

v, wPT 1M

mB1`D´pvq mB1

´D`pwq ntpv, wq ψ´pvqψ`pwq

“

ż

T 1Mψ´dσ`D´

ż

T 1Mψ`dσ´D` `Ope´κ

1tψ´` ψ``q .

Furthermore, if D´ and D` respectively have nonzero finite outer and inner skinningmeasures, and if pĂM,Γ, rF q satisfies the conditions of (1) or (2) above, then there exists κ2 ą 0such that, as tÑ `8,

ND´,D`, F ptq “σ`D´ σ

´

D`

δ mF eδ t

`

1`Ope´κ2tq

˘

.

The maps Op¨q depend on ĂM,Γ, F,D , and the speeds of mixing. The proof is a general-ization to nonzero potential of [PaP16c, Thm. 15].

Proof. We will follow the proofs of Theorem 11.1 and Corollary 12.3 to prove generalizationsof the assertions (1) and (2) by adding to these proofs a regularisation process of the testfunctions rφ˘η as for the deduction of [PaP14a, Theo. 20] from [PaP14a, Theo. 19]. We willthen deduce the last statement of Theorem 12.7 from these generalisations, again using thisregularisation process.

Let β be either α P s0, 1s small enough in the Hölder regularity case or ` P N big enoughin the Sobolev regularity case. We fix i P I´, j P I`, and we use the notation D˘, αγ , λγ , v˘γand rσ˘ of Equation (11.2). Let rψ˘ P C β

c pB1¯D

˘q. Under the assumptions of Assertion (1)or (2), we first prove the following avatar of Equation (11.4), indicating only the requiredchanges in its proof: there exists κ0 ą 0 (independent of rψ˘) such that, as T Ñ `8,

δ mF e´δ T

ÿ

γPΓ, 0ăλγďT

eş

αγrFrψ´pv´γ q

rψ`pv`γ q

“

ż

B1`D

´

rψ´ drσ`ż

B1´D

`

rψ` drσ´ `Ope´κ0T rψ´β rψ`βq . (12.4)

By [PaP16c, Lem. 6] and the Hölder regularity of the strong stable and unstable foliationsunder the assumptions of Assertion (1), or by the smoothness of the boundary of D˘ under

187 19/12/2016

the assumptions of Assertion (2), the maps f˘D¯

: V ˘η,RpB

1˘D

¯q Ñ B1˘D

¯ are respectivelyHölder-continuous or smooth fibrations, whose fiber over w P B1

˘D¯ is exactly V ˘w, η,R. By

applying leafwise the regularisation process described in the proof of [PaP14a, Theo. 20] tocharacteristic functions, there exist a constant κ1 ą 0 and χ˘η,R P C βpT 1

ĂMq such that‚ χ˘η,Rβ “ Opη´κ1q,‚ 1V ¯

η e´Opηq, R e´OpηqpB1¯D

˘qď χ˘η,R ď 1V ¯η,RpB

1¯D

˘q,

‚ for every w P B1¯D

˘, we haveż

V ¯w, η,R

χ˘η,R dν˘w “ ν˘w pV

¯w, η,Rq e

´Opηq “ ν˘w pV¯

w, η e´Opηq, R e´Opηqq eOpηq .

We now define the new test functions (compare with Section 10.1). For every w P B1¯D

˘,let

H˘η,Rpwq “1

ş

V ¯w, η,Rχ˘η,R dν

˘w.

Let Φ˘η : T 1ĂM Ñ R be the map defined by

Φ˘η “ pH˘η,R

rψ˘q ˝ f¯D˘

χ˘η,R .

The support of this map is contained in V ¯η,RpB

1¯D

˘q. Since M is compact in Assertion (1)and by homogeneity in Assertion (2), if R is large enough, by the definitions of the measuresν˘w , the denominator of H˘η,Rpwq is at least c η where c ą 0. The map H˘η,R is hence Hölder-continuous under the assumptions of Assertion (1), and it is smooth under the assumptionsof Assertion (2). Therefore Φ˘η P C βpT 1

ĂMq and there exists a constant κ2 ą 0 such that

Φ˘η β “ Opη´κ2 rψ˘βq .

As in Lemma 10.1, the functions Φ¯η are measurable, nonnegative and satisfyż

T 1ĂM

Φ¯η drmF “

ż

B1¯D

˘

rψ˘ drσ¯ .

As in the proof of Theorem 11.1, we will estimate in two ways the quantity

IηpT q “

ż T

0eδ t

ÿ

γPΓ

ż

T 1ĂMpΦ´η ˝ g

´t2q pΦ`η ˝ gt2 ˝ γ´1q drmF dt . (12.5)

We first apply the mixing property, now with exponential decay of correlations, as in Step2 of the proof of Theorem 11.1. For all t ě 0, let

Aηptq “ÿ

γPΓ

ż

vPT 1ĂM

Φ´η pg´t2vq Φ`η pg

t2γ´1vq drmF pvq .

Then with κ ą 0 as in the definitions of the exponential mixing for the Hölder or Sobolevregularity, we have

Aηptq “1

mF

ż

T 1ĂM

Φ´η drmF

ż

T 1ĂM

Φ`η drmF ` O`

e´κ tΦ´η βΦ`η β

˘

“1

mF

ż

B1`D

´

rψ´ drσ`ż

B1´D

`

rψ` drσ´ ` O`

e´κ tη´2κ2 rψ´β rψ`β

˘

.

188 19/12/2016

Hence by integrating,

IηpT q “eδ T

δ mF

´

ż

B1`D

´

rψ´ drσ`ż

B1´D

`

rψ` drσ´ ` O`

e´κT η´2κ2 rψ´β rψ`β

˘

¯

. (12.6)

Now, as in Step 3 of the proof of Theorem 11.1, we exchange the integral over t and thesummation over γ in the definition of IηpT q, and we estimate the integral term independentlyof γ:

IηpT q “ÿ

γPΓ

ż T

0eδ t

ż

T 1ĂMpΦ´η ˝ g

´t2q pΦ`η ˝ gt2 ˝ γ´1q drmF dt .

Let pΦ˘η “ H˘η,R ˝ f¯

D˘χ˘η,R, so that Φ˘η “

rψ˘ ˝ f¯D˘

pΦ˘η . By the last two properties of theregularised maps χ˘η,R, we have, with φ¯η defined as in Equation (10.4),

φ˘η e´Opηq, R e´Opηq, B1

¯D˘ e

´Opηq ď pΦ˘η ď φ˘η eOpηq . (12.7)

If v P T 1ĂM belongs to the support of pΦ´η ˝ g´t2q pΦ`η ˝ gt2 ˝ γ´1q, then we have v P

gt2V `η,RpB

1`D

´q X g´t2V ´η,RpγB

1´D

`q. Hence the properties (i), (ii) and (iii) of Step 3M ofthe proof of Theorem 11.1 still hold (with Ω´ “ B

1`D

´ and Ω` “ B1´pγD

`q). In particular,if w´ “ f`

D´pvq and w` “ f´

γD`pvq, we have, by Assertion (iii) in Step 3M of the proof of

Theorem 11.1,6 thatdpw˘, v˘γ q “ Opη ` e´λγ2q .

Hence, with κ3 “ α in the Hölder case and κ3 “ 1 in the Sobolev case (we may assume that` ě 1), we have

| rψ˘pw˘q ´ rψ˘pv˘γ q | “ Oppη ` e´λγ2qκ3 rψ˘βq .

Therefore there exists a constant κ4 ą 0 such that

IηpT q “ÿ

γPΓ

`

rψ´pv´γ qrψ`pv`γ q `Oppη ` e´λγ2qκ4 rψ´β rψ

`βq˘

ˆ

ż T

0eδ t

ż

vPT 1ĂM

pΦ´η pg´t2vq pΦ`η pγ

´1gt2vq drmF pvq dt .

Now, using the inequalities (12.7), Equation (12.4) follows as in Steps 3M and 4M of theproof of Theorem 11.1, by taking η “ e´κ5T for some κ5 ą 0 and using the effective controlgiven by Equation (11.19) in Step 4M.

In order to prove Assertions (1) and (2) of Theorem 12.7, we may assume that the supportsof ψ˘ are small enough, say contained in Bpx˘, εq for some x˘ P T 1M and ε small enough. Letrx˘ be lifts of x˘ and let rψ˘ P C β

c pT 1ĂMq with support in Bp rx˘, εq be such that rψ˘ “ ψ˘ ˝Tp

where p : ĂM Ñ M is the universal cover. By a finite summation argument and Equation(12.4), we have

δ mF e´δ T

ÿ

iPI´„, jPI`„, γPΓ0ăλi, γjďT

eş

αγrFrψ´pv´γ q

rψ`pv`γ q

“

ż

B1`D

´

rψ´ drσ`ż

B1´D

`

rψ` drσ´ `Ope´κ0T rψ´β rψ`βq . (12.8)

6See also the picture at the beginning of the proof of Lemma 11.7.

189 19/12/2016

Assertions (1) and (2) are deduced from this equation in the same way that Corollary 12.3is deduced from Theorem 11.1. Taking the functions ψ˘k to be the constant functions 1 inAssertion (1) gives the last statement of Theorem 12.7 under the assumptions of Assertion(1). An approximation argument gives the result under the assumptions of Assertion (2). l

12.4 Equidistribution and counting for quotient simplicial andmetric trees

In this Section, we assume that X is the geometric realisation of a locally finite metrictree without terminal vertices pX, λq, and that Γ is a (nonelementary discrete) subgroupof AutpX, λq. Let rc : EX Ñ R be a system of conductances for Γ, and let c : ΓzEX Ñ Rbe its quotient function. We assume in this Section that the potential rF is the potential rFcassociated7 with c. Let δc “ δΓ, Fc be the critical exponent of pΓ, Fcq and let rmc “ rmFc andmc “ mFc be the Gibbs measures of Fc for the continuous time geodesic flow on respectivelyGX and ΓzGX, as well as for the discrete time geodesic flow on respectively GX and ΓzGXwhen pX, λq is simplicial, that is, if λ is constant with value 1.

Let D˘ be simplicial subtrees of X, with the edge length map induced by λ,8 such thatthe Γ-equivariant families D˘ “ pγD˘qγPΓΓD˘

are locally finite in X.9

For all γ, γ1 in Γ such that γD´ and γ1D` are disjoint, we denote by αγ, γ1 the com-mon perpendicular from γD´ to γ1D` (which is an edge path in X), with length λγ, γ1 “

dpγD´, γ1D`q P N, and by α˘γ, γ1 Pp

G X its parametrisations as in the beginning of Chapter11: it is the unique map from R to X such that α´γ, γ1ptq P γV D´ is the origin opαγ, γ1q ofthe edge path αγ, γ1 if t ď 0, α´γ, γ1ptq P γ

1V D` is the endpoint tpαγ, γ1q of the edge path αγ, γ1if t ě λγ, γ1 , and α´γ, γ1 |r0, λγ, γ1 s

is the shortest geodesic arc starting from a point of γD´ and

ending at a point of γ1D`.For all γ, γ1 in Γ such that γD´ and γ1D` are disjoint, we define the multiplicity of the

common perpendicular αγ, γ1 from γD´ to γ1D` as

mγ, γ1 “1

CardpγΓD´γ´1 X γ1ΓD`γ

1´1q. (12.9)

Note that mγ, γ1 “ 1 for all γ, γ1 P Γ when Γ acts freely on EX (for instance when Γ istorsion-free). Generalising the definition for simplicial trees in Section 11.4, we set

cpαq “kÿ

i“1

cpeiqλpeiq ,

for any edge path α “ pe1, . . . , ekq.For n P N´ t0u, let

ND´,D`pnq “ÿ

rγsPΓD´zΓΓD`0ădpD´,γD`qďn

me, γ ecpαe, γq ,

7See Section 3.5.8By abuse, we will still denote by D˘ the geometric realisation |D˘|λ.9We leave to the reader the extension to more general locally finite families of subtrees, as for instance

finite unions of those above.

190 19/12/2016

where Γ acts diagonally on pΓΓD´q ˆ pΓΓD`q and d is the the distance on X “ |X|λ. WhenΓ is torsion-free, ND´,D`pnq is the number of edge paths in the graph ΓzX of length at mostn, starting by an outgoing edge from the image of D´ and ending by the opposite of anoutgoing edge from the image of D`, with multiplicities coming from the fact that ΓD˘zD˘is not assumed to be embedded in ΓzX, and with weights coming from the conductances.

In the next results, we distinguish the continuous time case (Theorem 12.8) from thediscrete time case (Theorem 12.9). We leave to the reader the versions without the assumptionδc ą 0, giving for every τ P N´ t0u an asymptotic on

ND´,D`, τ pnq “ÿ

rγsPΓD´zΓΓD`n´τădpD´,γD`qďn

me, γ ecpαe, γq .

When ΓzX is compact, c “ 0 and D˘ are reduced to points, the counting results inTheorems 12.8 and 12.9 are proved in [Gui]. When D˘ are singletons, Theorem 12.8 is dueto [Rob2] if c “ 0. Otherwise, the result seems to be new.

Theorem 12.8. Let pX, λq, Γ, D˘ and c be as in the beginning of this Section. Assume thatthe critical exponent δc is finite and positive, that the skinning measures σ˘D¯ are finite andnonzero, and that the Gibbs measure mc is finite and mixing for the continuous time geodesicflow. Then as tÑ `8, the measures

δc mc e´δc t

ÿ

rγs PΓD´zΓΓD`0ădpD´,γ D`qďt

me, γ ecpαe, γq ∆Γα´e,γ

b∆Γα`γ´1,e

narrow converge to σ`D´ b σ´

D` in Γz

p

GX ˆ Γz

p

GX, and

ND´,D`ptq „σ`D´ σ

´

D`

δc mceδc t .

Proof. By Theorem 11.1, we have

limtÑ`8

δc mc e´δc t

ÿ

pa,b,γqPΓΓD´ˆΓΓD`ˆΓ

0ădpaD´, γbD`qďt

e

ş

αa,γbrFc

∆α´a,γbb∆α`

γ´1a,b

“ rσ`D´ b rσ´D` ,

not only for the weak-star convergence onp

G X ˆp

G X, but also by Step 1 of the proof ofTheorem 11.1, for the narrow convergence, as σ`D´ and σ´D` are finite. Recall that given adiscrete group G acting properly (but not necessarily freely) on a locally compact space Z,the induced measure on GzZ of a (positive, Radon) measure µ on Z is a measure µ whichdepends linearly and continuously (both for the weak-star and narrow topologies) on µ, andsatisfies ∆z “

1|Gz |

∆Gz for every z P Z. See for instance Section [PauPS, §2.4] for moredetails.

The group Γˆ Γ acts on ΓΓD´ ˆ ΓΓD` ˆ Γ by

pa1, b1q ¨ pa, b, γq “ pa1a, b1b, a1γpb1q´1q .

and the map from the discrete set ΓΓD´ ˆ ΓΓD` ˆ Γ top

G Xˆp

G X which sends pa, b, γq topα´a,γb , α

`

γ´1a,bq is pΓˆΓq-equivariant. In particular, the pushforward of measures by this map

sends the unit Dirac mass at pa, b, γq to ∆α´a,γbb∆α`

γ´1a,b

.

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Every orbit of ΓˆΓ on ΓΓD´ˆΓΓD`ˆΓ has a representative of the form pΓD´ ,ΓD` , γq forsome γ P Γ, since pa, bq¨pΓD´ ,ΓD` , a

´1γ bq “ paΓD´ , bΓD´ , γq. Furthermore the double class inΓD´zΓΓD` of such a γ is uniquely defined, and the stabiliser of pΓD´ ,ΓD` , γq has cardinality|ΓD´ X γΓD`γ

´1|, since pa, bq ¨ pΓD´ ,ΓD` , γq “ pΓD´ ,ΓD` , γ1q if and only if a P ΓD´ , b P ΓD`

and aγ b´1 “ γ1. When γ1 “ γ, this happens if and only if b “ γ´1aγ and a P ΓD´XγΓD`γ´1.

Hence the measures

δc mc e´δc t

ÿ

rγs PΓD´zΓΓD`0ădpD´,γ D`qďt

1

|ΓD´ X γΓD`γ´1|

eş

Γαe,γFc ∆Γα´e,γ

b∆Γα`γ´1,e

narrow converge as tÑ `8 to σ`D´ bσ´

D` in Γz

p

G X ˆΓz

p

G X. By applying this convergenceto the constant function 1, and by the finiteness and nonvanishing of σ`D´ and σ´D` , the resultfollows using the defining property of the potential Fc, see Proposition 3.11. l

In the remainder of this Section, we consider simplicial trees with the discrete time geodesicflow.

Theorem 12.9. Let pX, λq, Γ, rc and D˘ be as in the beginning of this Section, with λ constantwith value 1. Assume that the critical exponent δc is finite and positive. If the Gibbs measuremc is finite and mixing for the discrete time geodesic flow and the skinning measures σ˘D¯ arefinite and nonzero, then as nÑ `8, the measures

eδc ´ 1

eδcmc e

´δc nÿ

rγs PΓD´zΓΓD`0ădpD´,γ D`qďn

me, γ ecpαe, γq ∆Γα´e,γ

b∆Γα`γ´1,e

narrow converge to σ`D´ b σ´

D` in Γz

p

G Xˆ Γz

p

G X and

ND´,D`pnq „eδc σ`D´ σ

´

D`

peδc ´ 1q mceδc n .

Proof. The claims follow as in Theorem 12.8, replacing Theorem 11.1 by Theorem 11.8. l

Examples 12.10. (1) Let X,Γ, c be as in Theorem 12.9, and let D´ “ txu and D` “ tyu forsome x, y P V X. If the Gibbs measure mc is finite and mixing for the discrete time geodesicflow, then we have a version of Roblin’s simultaneous equidistribution theorem with potential,see Corollary 11.2, and the number Nx, ypnq of nonbacktracking edge paths of length at mostn from x to y (counted with weights and multiplicities) satisfies

Nx, ypnq „eδc µ`x µ

´y

peδc ´ 1q mc |Γx| |Γy|eδc n .

(2) If Y is a finite connected nonbipartite pq ` 1q-regular graph (with q ě 2) and Y˘ arepoints, then the number of nonbacktracking edge paths from Y´ to Y` of length at most nis equivalent as nÑ `8 to

q ` 1

q ´ 1

qn

|V Y|`Oprnq (12.10)

for some r ă q. Indeed, by Theorem 12.9 with X the universal cover of Y, Γ its covering groupand c “ 0, we have δc “ ln q and mc is the Bowen-Margulis measure, so that normalizing the

192 19/12/2016

Patterson measures to be probability measures, we have mc “qq`1 |V Y| by Equation (8.3).

We refer to Section 12.6 (see Remark (i) following Theorem 12.17) for the error term.Let Y the figure 8-graph with a single vertex and four directed edges, and let Y˘ be the

the singleton consisting of its vertex. In this simple example, it is easy to count by handthat the number of loops of length exactly n without backtracking in Y is 4 3n´1. Thus thenumber N pnq of common perpendiculars of the vertex to itself of length at most n is by asimple geometric sum 2p3n ´ 1q. This agrees with Equation (12.10) that gives N pnq „ 2 3n

as nÑ `8.(3) Let Y be a finite connected nonbipartite pq ` 1q-regular graph (with q ě 2). Let Y˘ beregular connected subgraphs of degrees q˘ ě 0. Then the number N pnq of edge paths oflength at most n starting transversally to Y´ and ending transversally to Y` satisfies

N pnq “pq ` 1´ q´qpq ` 1´ q`q |V Y´| |V Y`|

pq2 ´ 1q |V Y|qn `Oprnq

for some r ă q. This is a direct consequence of Theorem 12.9, using Proposition 8.1 (3) andProposition 8.4 (3), and refering to Section 12.6 (see Remark (i) following Theorem 12.17) forthe error term.

We refer for instance to Section 15.2 for examples of counting results in graphs of groupswhere the underlying graph is infinite.

Remark 12.11. A common perpendicular in a simplicial tree is, in the language of graphtheory, a non-backtracking walk. Among other applications (when restricting to groups Γacting freely, which is never the case if Γ is a nonuniform lattice in the tree X, that is, when thequotient graph of groups ΓzX is infinite but has finite volume), Theorem 12.9 gives a completeasymptotic solution to the problem of counting non-backtracking walks from a given vertex toa given vertex of a (finite) nonbipartite graph. See Theorem 12.12 for the corresponding resultin bipartite graphs and for example [ABLS, Th. 1.1], [AnFH, p. 4290,4302], [Fri2, L. 2.3], [Sod,Prop. 6.4] for related results. Anticipating on the error terms that we will give in Section12.6, note that the paper [ABLS, Th. 1.1] for instance gives a precise speed using spectralproperties, more precise than the ones we obtain.

In some applications (see the examples at the end of this Section), we encounter bipartitesimplicial graphs and, consequently, their discrete time geodesic flow is not mixing. Thefollowing result applies in this context.

Until the end of this section, we assume that the simplicial tree X has a Γ-invariantstructure of a bipartite graph, and we denote by V X “ V1X\V2X the corresponding partitionof its set of vertices. For every i P t1, 2u, we denote by

p

GiX the space of generalised discretegeodesic lines ` P

p

G X such that `p0q P ViX, so that we have a partitionp

G X “p

G1 X \p

G2 X.Note that if the basepoint x0 lies in ViX, then GevenX is equal to

p

GiXXGX. For all i, j P t1, 2u,we define

ND´,D`, i, jpnq “ÿ

rγsPΓD´zΓΓD`0ădpD´,γD`qďn

opαe,γqPViX, tpαe,γqPVjX

me, γ ecpαe, γq .

Theorem 12.12. Let pX, λq, Γ and c be as in the beginning of this Section, with λ constantwith value 1. Assume that the critical exponent δc is finite and positive. If X has a Γ-invariant

193 19/12/2016

structure of a bipartite graph as above, if the restriction to ΓzGevenX of the Gibbs measure mc

is finite and mixing for the square of the discrete time geodesic flow, then for all i, j P t1, 2usuch that the measures σ´D´, i and σ

`

D`, j are finite and nonzero, as n tends to `8 with n ” i´jmod 2, the measures

e2 δc ´ 1

2 e2 δcmc e

´δc nÿ

rγs PΓD´zΓΓD`0ădpD´,γ D`qďn

opαe,γqPViX, tpαe,γqPVjX

me, γ ecpαe, γq ∆Γα´e,γ

b∆Γα`γ´1,e

narrow converge to σ`D´, i b σ´

D`, j in Γz

p

G Xˆ Γz

p

G X and

ND´,D`, i, jpnq „2 e2 δc σ`D´, i σ

´

D`, j

pe2 δc ´ 1q mceδc n .

Proof. This Theorem is proved in the same way as the above Theorem 12.9 using Theorem11.10. Note that we have a pΓˆ Γq-invariant partition

p

G Xˆp

G X “ğ

pi, jqPt1, 2u2

p

GiXˆp

Gj X ,

that α´e, γ Pp

GiX if and only if opαe, γq P ViX, and that α`γ´1, e

P

p

Gj X if and only if tpαe, γq P VjX,since α`

γ´1, ep0q “ γ´1α`e, γp0q “ γ´1tpαe, γq. l

Examples 12.13. (1) Let X,Γ, c be as in Theorem 12.12, and let D´ “ txu and D` “ tyufor some vertices x, y in the same ViX for i P t1, 2u. If the restriction to ΓzGevenX of theGibbs measure mc is finite and mixing for the square of the discrete time geodesic flow, thenas nÑ `8 is even,

ND´,D`pnq „2 e2 δc

e2 δc ´ 1

µ`x µ´y

mc |Γx| |Γy|eδc n .

Indeed, we have ND´,D`pnq “ ND´,D`, i, ipnq and σ˘

D˘, i “ σ˘D˘ .

(2) Let Y be the complete biregular graph with q`1 vertices of degree p`1 and p`1 verticesof degree q ` 1. Let Y˘ “ tyu be a fixed vertex of degree p` 1. Note that Y being bipartite,all common perpendiculars from y to y have even length, (the shortest one having length 4).Then as n is even and tends to `8, we have

NY´,Y`pnq „qpp` 1q

pq ` 1qppq ´ 1qppqqn2 .

Indeed, the biregular tree Xp,q of degrees pp` 1, q` 1q is a universal cover of Y with coveringgroup Γ acting freely and cocompactly, so that with c “ 0 we have δc “ ln

?pq and the Gibbs

measure mc is the Bowen-Margulis measure mBM. If we normalise the Patterson density suchthat µy “ p`1

?p , then by Proposition 8.1 (2), we have mBM “ 2pp ` 1qpq ` 1q. Thus the

result follows from Example (1). Note that if p “ q, then

NY´,Y`pnq „q

q2 ´ 1qn ,

and the constant in front of qn is indeed different from that in the nonbipartite case.

194 19/12/2016

(3) Let Y be a finite biregular graph with vertices of degrees p` 1 and q ` 1, where p, q ě 2,and let V Y “ VpY\ VqY be the corresponding partition. If Y´ “ tvu where v P VpY and Y`is a cycle of length L ě 2, then as N Ñ `8, the number of common perpendiculars of evenlength at most 2N from Y´ to Y` is equivalent to

L q pp´ 1q

2 ppq ´ 1q |VpY|ppqqN ,

and the number of common perpendiculars of odd length at most 2N ´ 1 from v to Y` isequivalent to

L pq ´ 1q

2 ppq ´ 1q |VpY|ppqqN .

Proof. The cycle Y` has even length L and has L2 vertices in both VpY and VqY. A common

perpendicular from Y´ to Y` has even length if and only if it ends at a vertex in VpY.Let XÑ Y be a universal cover of Y, whose covering group Γ acts freely and cocompactly

on X. Let D´ “ trvu where rv P VpX is a lift of v, and let D´ be a geodesic line in X mapping toY`. We use Theorem 12.12 with V1X the (full) preimage of VpY in X, with V2X the premiageof VqY in X and with c “ 0, so that δc “ ln

?pq and mc “ mBM. Let us normalise the

Patterson density of Γ as in Proposition 8.1 (2), so that

σ`D´, 1 “ µ rv “p` 1?p

.

The mass for the skinning measure of the part of the inner unit normal bundle of Y` withbasepoint in VpY is (see Corollary 8.5)

L

2

p` 1?p

p´ 1

p` 1“Lpp´ 1q

2?p

and its complement has mass Lpq´1q2?q . Recall also that, by Proposition 8.1 (2), considering the

graph Y as a graph of groups with trivial groups,

mBM “ TVolpYq “ |EY| “ 2pp` 1q|VpY| “ 2pq ` 1q|VqY| .

The claim about the common perpendiculars of even length at most 2N follows fromTheorem 12.12 with i “ j “ 1, since

2 e2 δc σ`D´, i σ´

D`, j

pe2 δc ´ 1q mc“

2 pq p`1?p

Lpp´1q2?p

ppq ´ 1q 2 pp` 1q |VpY|“

L q pp´ 1q

2 ppq ´ 1q |VpY|.

The claim about the common perpendiculars of odd length at most 2N´1 follows similarlyfrom Theorem 12.12 with i “ 1 and j “ 2. l

(4) Let Y be a finite biregular graph with vertices of degrees p` 1 and q ` 1, where p, q ě 2,and let V Y “ VpY \ VqY be the corresponding partition. If Y´ and Y` are cycles of lengthL´ ě 2 and L` ě 2 respectively, then as N Ñ `8, the number of common perpendicularsof length at most N from Y´ to Y` is equal to

p?q `

?pq2 L´ L`

2 ppq ´ 1q CardpEYqp?pqqN`2 `OprN q (12.11)

195 19/12/2016

for some r ă ?pq.

Proof. As in the above proof of Example (3), let X Ñ Y be a universal cover of Y, withcovering group Γ and let D˘ be a geodesic line in X mapping to Y˘. We normalise thePatterson density pµxqxPV X of Γ so that µx “

degXpxq?degXpxq´1

. By Proposition 8.4 (3) with

k “ 1 and trivial vertex stabilisers, and since a simple cycle of length λ in a biregular graphof different degrees p` 1 and q ` 1 has exactly λ

2 vertices of degree either p` 1 or q ` 1, wehave

σ˘D¯ “ÿ

ΓxPY¯

µx pdegXpxq ´ kq

degXpxq“

ÿ

yPVpY¯

?p`

ÿ

yPVqY¯

?q “ L¯

?p`

?q

2.

The result without the error term then follows from Theorem 12.12, using Proposition 8.1 (2)and Remark 2.11.

We refer to Section 13.2 (see Remark (ii) following Theorem 12.17) for the error term. l

Remark 12.14. If Y is a finite bipartite pp ` 1q-regular graph, Y´ consists in a vertex andY` is a cycle of length L, then Example (3) above gives

NY´, Y`pnq „L p pp´ 1q

2 pp2 ´ 1q |V Y|pn `

L pp´ 1q

2 pp2 ´ 1q |V Y|pn “

L

|V Y|pn

for the number NY´, Y`pnq of common perpendiculars from Y´ to Y` with length at most n.This is the same result as for nonbipartite trees.

12.5 Counting for simplicial graphs of groups

In this Section, we give an intrinsic translation “a la Bass-Serre” of the counting result inTheorem 12.9 using graphs of groups (see [Ser3] and Section 2.7 for background information).

Let pY, G˚q be a locally finite, connected graph of finite groups, and let pY˘, G˘˚ q beconnected subgraphs of subgroups.10 Let c : EYÑ R be a system of conductances on Y.

Let X be the Bass-Serre tree of the graph of groups pY, G˚q (with geometric realisationX “ |X|1) and Γ its fundamental group (for an indifferent choice of basepoint). Assume thatΓ is nonelementary. We denote by G pY, G˚q “ ΓzGX and

`

gt : G pY, G˚q Ñ G pY, G˚q˘

tPZ thequotient of the (discrete time) geodesic flow on GX, by rc : X Ñ R the (Γ-invariant) lift ofc, with δc its critical exponent and rFc : T 1X Ñ R its associated potential, by mc the Gibbsmeasure on G pY, G˚q associated with a choice of Patterson densities pµ˘x qxPX for the pairspΓ, F˘c q, by D˘ two subtrees in X such that the quotient graphs of groups ΓD˘zzD˘ identifywith pY˘, G˘˚ q (see below for precisions), and by σ˘

pY¯,G¯˚ qthe associated skinning measures.

The fundamental groupoid πpY, G˚q of pY, G˚q11 is the quotient of the free product of thegroups Gv for v P V Y and of the free group on EY by the normal subgroup generated by theelements e e and e ρepgq e ρ epgq´1 for all e P EY and g P Ge. We identify each Gx for x P V Ywith its image in πpY, G˚q.

Let n P N ´ t0u. A (locally) geodesic path of length n in the graph of groups pY, G˚q isthe image α in πpY, G˚q of a word, called reduced in [Bass, 1.7],

h0 e1 h1 e2 . . . hn´1 en hn10See Section 2.7 for definitions and background.11denoted by F pY, G˚q in [Ser3, §5.1], called the path group in [Bass, 1.5], see also [Hig]

196 19/12/2016

with

‚ ei P EY and tpeiq “ opei`1q for 1 ď i ď n ´ 1 (so that pe1, . . . , enq is an edge path inthe graph Y;

‚ h0 P Gope1q and hi P Gtpeiq for 1 ď i ď n;

‚ if ei`1 “ ei then hi does not belong to ρeipGeiq, for 1 ď i ď n´ 1.

Its origin is opαq “ ope1q and its endpoint is tpαq “ tpenq. They do not depend on the chosenwords with image α in πpY, G˚q.

A common perpendicular of length n from pY´, G´˚ q to pY`, G`˚ q in the graph of groupspY, G˚q is the double coset

rαs “ G´opαq α G`

tpαq

of a geodesic path α of length n as above, such that:‚ α starts transversally from pY´, G´˚ q, that is, its origin opαq “ ope1q belongs to V Y´

and h0 R G´

ope1qρ e1pGe1q if e1 P EY´,

‚ α ends transversally in pY`, G`˚ q, that is, its endpoint tpαq “ tpenq belongs to Y` andhn R ρenpGenqG

`

tpenqif en P EY`.

Note that these two notions do not depend on the representative of the double cosetG´opαq α G

`

tpαq, and we also say that the double coset rαs starts transversally from pY´, G´˚ q orends transversally in pY´, G´˚ q.

We denote by PerpppY˘, G˘˚ q, nq the set of common perpendiculars in pY, G˚q of length nfrom pY´, G´˚ q to pY`, G`˚ q. We denote by

cpαq “nÿ

i“1

cpeiq

the conductance of a geodesic path α as above, which depends only on the double class rαs.We define the multiplicity mα of a geodesic path α as above by

mα “1

CardpG´opαq X αG`

tpαq α´1q

.

It depends only on the double class rαs of α. We define the counting function of the commonperpendiculars in pY, G˚q of length at most n from pY´, G´˚ q to pY`, G`˚ q (counted withmultiplicities and with weights given by the system of conductances c) as

NpY´,G´˚ q, pY`,G

`˚ qpnq “

ÿ

rαsPPerpppY˘,G˘˚ q,nq

mα ecpαq .

Theorem 12.15. Let pY, G˚q, pY˘, G˘˚ q and c be as in the beginning of this Section. Assumethat the critical exponent δc of c is positive and that the Gibbs measure mc on G pY, G˚q is finiteand mixing for the discrete time geodesic flow. Assume that the skinning measures σ˘

pY¯,G¯˚ qare finite and nonzero. Then as n P N tends to 8

NpY´,G´˚ q, pY`,G

`˚ qpnq „

eδc σ`pY´,G´˚ q

σ´pY`,G`˚ q

peδc ´ 1q mceδc n .

197 19/12/2016

Proof. Let X be the Bass-Serre tree of pY, G˚q and Γ its fundamental group (for an indifferentchoice of basepoint). As seen in Section 2.7, the Bass-Serre trees D˘ of pY˘, G˘˚ q, withfundamental groups Γ˘, identify with simplicial subtrees D˘ of X, such that Γ˘ are thestabilisers ΓD˘ of D˘ in Γ. In particular, the maps pΓD˘zD˘q Ñ pΓzXq induced by theinclusion maps D˘ Ñ X by taking quotient, are injective:

@ γ P Γ, @ z P V D˘ Y ED˘, if γz P V D˘ Y ED˘, then D γ˘ P ΓD˘ , γ˘z “ γz . (12.12)

As in Definition 2.10, for all z P V Y Y EY and e P EY, we fix a lift rz P V X Y EX of zand ge P Γ, such that re “ re, ge Ątpeq “ tpreq, Gz “ Γ

rz, and the monomorphism ρe : Ge Ñ Gtpeqis γ ÞÑ g´1

e γge. We assume, as we may, that rz P V D˘ Y ED˘ if z P V Y˘ Y EY˘. Weassume, as we may using Equation (12.12), that if e P EY˘, then ge P ΓD˘ . We denote byp : XÑ Y “ ΓzX the canonical projection.

fk

γk

f1

re1

γ1

Ąei`1

gei

Y´g0

e1 giei

Ćtpeiq

fi fi`1

γi`1

gei`1

rei

γi

Y`

ek

gk

γ1γk`1γ1´1

γD´

γγ0γ´1

γ Ćope1q γ1 Ćtpekq

γ1D`

tpfkqopf1q

rek

γ1

gekĆtpekq

γ

ge1Ćope1q

ei`1

For all γ, γ1 P Γ such that γD´ and γ1D` are disjoint, the common perpendicular αγD´, γ1D`from γD´ to γ1D` is an edge path pf1, f2, . . . , fkq with opf1q P γD´ and tpfkq P γ1D`. Notethat γ´1 opf1q and ppopf1qq„

are two vertices of D´ in the same Γ-orbit, and that γ1´1 tpfkq andpptpfkqq„

are two vertices of D` in the same Γ-orbit. Hence by Equation (12.12), we may chooseγ0 P ΓD´ such that γ0γ

´1 opf1q “ ppopf1qq„

and γk`1 P ΓD´ such that γk`1γ1´1 tpfkq “ pptpfkqq„

.For 1 ď i ď k, choose γi P Γ such that γifi “ Ćppfiq. We define

‚ ei “ ppfiq for 1 ď i ď k,

‚ hi “ g´1ei γiγi`1

´1g ei`1 , which belongs to ΓĆtpeiq

“ Gtpeiq for 1 ď i ď k ´ 1,

‚ h0 “ γ0γ´1γ´1

1 g e1 “ γ´1pγγ0γ´1qγ´1

1 g e1 , which belongs to ΓČope1q

“ Gope1q,

‚ hk “ g´1ekγkγ

1γk`1´1 “ g´1

ekγkpγ

1γk`1γ1´1q´1γ1, which belongs to Γ

Ćtpekq“ Gtpekq.

Lemma 12.16. (1) The word h0e1h1 . . . hk´1ekhk is reduced. Its image α in the fundamen-tal groupoid πpY, G˚q does not depend on the choices of γ1, . . . , γk, and starts transver-sally from pY´, G´˚ q and ends transversally in pY`, G`˚ q. The double class rαs of α isindependent of the choices of γ0 and γk`1.

198 19/12/2016

(2) The map rΘ from the set of common perpendiculars in X between disjoint images of D´and D` under elements of Γ, into the set of common perpendiculars in pY, G˚q frompY´, G´˚ q to pY`, G`˚ q, sending αγD´, γ1D` to rαs, is constant under the action of Γ atthe source, and preserves the lengths and the multiplicities.

(3) The map Θ induced by rΘ from the set of Γ-orbits of common perpendiculars in X betweendisjoint images of D´ and D` under elements of Γ into the set of common perpendicularsin pY, G˚q from pY´, G´˚ q to pY`, G`˚ q is a bijection, preserving the lengths and themultiplicities.

Proof. (1) If ei`1 “ ei, then by the definition of hi, we have

hi P ρeipGeiq “ g´1ei Γ

reigei ðñ gei hi g´1ei rei “ rei

ðñ gei g´1ei γiγi`1

´1g ei`1 g´1ei rei “ rei

ðñ γiγi`1´1

Ąei`1 “ rei

ðñ γi`1´1

Ąei`1 “ γ´1i rei ðñ fi`1 “ fi .

Hence the word h0e1h1 . . . hk´1ekhk is reduced.The element γi for i P t1, . . . , ku is uniquely determined up to multiplication on the left

by an element of Γrei “ Gei . If we fix12 i P t1, . . . , ku and if we replace γi by γ1i “ αγi for some

α P Gei , then only the elements hi´1 and hi change, replaced by elements that we denote byh1i´1 and h1i respectively. We have (if 2 ď i ď k´ 1, but otherwise the argument is similar bythe definitions of h0 and hk)

h1i´1 ei h1i “ g´1

ei´1γi´1 γi

´1α´1g ei ei g´1ei αγi γi`1

´1g ei`1

“ g´1ei´1

γi´1 γi´1g ei ρ eipαq

´1 ei ρeipαq g´1ei γi γi`1

´1g ei`1 .

Since ρ eipαq´1 ei ρeipαq is equal to ei ´1 “ ei in the fundamental groupoid, the words h1i´1 ei h1i

and hi´1 ei hi have the same image in πpY, G˚q. Therefore α is independent on the choices ofγ1, . . . , γk.

We have opαq “ ope1q P V Y´ and tpαq “ tpekq P V Y`, hence α starts from Y´ and endsin Y`.

Assume that e1 P EY´. Let us provethat h0 P G

´

ope1qρ e1pGe1q if and only if

f1 P γ ED´.

γ´1f1

Ćope1q

γ0g e1

opγ´1f1q

re1 D´

By the definition of ρ e1 , we have h0 P G´ope1q ρ e1pGe1q if and only if there exists α P

ΓČope1q

XΓD´ such that α´1 h0 P g´1e1

ΓĂe1 g e1 . By the definition of h0 and since γ1 maps f1 to

re1, we have

α´1 h0 P g´1e1

ΓĂe1 g e1 ðñ g e1 α

´1`

γ0γ´1γ´1

1 g e1˘

g ´1e1

re1 “ re1

ðñ f1 “ γ γ´10 α g ´1

e1re1 .

12We leave to the reader the verification that the changes induced by various i’s do not overlap.

199 19/12/2016

Since re1 P ED´ and γ0, α, g e1 all belong to ΓD´ , this last condition implies that f1 P γ ED´.Conversely (for future use), if f1 P γ ED´, then (see the above picture) γ0 γ

´1f1 is an edgeof D´ with origin Ćope1q, in the same orbit that the edge g ´1

e1re1 of D´, which also has

origin Ćope1q. By Equation (12.12), this implies that there exists α P ΓČope1q

X ΓD´ such that

f1 “ γ γ´10 α g ´1

e1re1. By the above equivalences, we hence have that h0 P G

´

ope1qρ e1pGe1q.

Similarly, one proves that if ek P EY`, then hk P ρekpGekqG`

tpekqif and only if fk P γ1ED`.

Since pf1, . . . , fnq is the common perpendicular edge path from γ D´ to γ1D`, this proves thatα starts transversally from Y´ and ends transversally in Y´.

Note that the element γ0 P ΓD´ is uniquely defined up to multiplication on the left byan element of Γ

Čope1qX ΓD´ “ G´ope1q, and appears only as the first letter in the expression of

h0. Note that the element γk`1 P ΓD` is uniquely defined up to multiplication on the left byan element of Γ

ĆtpekqX ΓD` “ G`tpekq, hence γ

´1k`1 is uniquely defined up to multiplication on

the right by an element of G`tpekq, and appears only as the last letter in the expression of hk.Therefore α is uniquely defined in the fundamental groupoid πpY, G˚q up to multiplicationon the left by an element of G´ope1q and multiplication on the right by an element of G`tpekq,that is, the double class rαs Ă πpY, G˚q is uniquely defined.

(2) Let β be an element in Γ and let x “ αγD´, γ1D` be a common perpendicular in Xbetween disjoint images of D´ and D` under elements of Γ. Let us prove that rΘpβ xq “ rΘpxq.

Since β x “ αβ γD´, β γ1D` , in the construction of rΘpβ xq, we may take, instead of theelements γ0, γ1, . . . , γk, γk`1 used to construct rΘpxq, the elements

γ70 “ γ0, γ71 “ γ1 β´1, . . . , γ7k “ γk β

´1, γ7k`1 “ γk`1.

And instead of γ and γ1, we now may use γ7 “ β γ and γ17 “ β γ1.The only terms involving γ, γ1, γ1, . . . , γk in the construction of rΘpxq come under the form

γ´1γ´11 in h0, γi γi`1

´1 in hi for 1 ď i ď k´ 1, and γkγ1 in hk. Since pγ7q´1pγ71q´1 “ γ´1γ´1

1 ,pγ7i qpγ

7

i`1q´1 “ γi γi`1

´1 for 1 ď i ď k ´ 1, and pγ7kqpγ17q “ γkγ

1, this proves that rΘpβ xq “rΘpxq, as wanted.

It is immediate that if the length of αγD´, γ1D` is k, then the length of rαs is k.Let us prove that the multiplicity, given in Equation (12.9),

mγ, γ1 “1

CardpγΓD´γ´1 X γ1ΓD`γ

1´1q

of the common perpendicular αγD´, γ1D` in X between γ D´ and γ1D` is equal to the multi-plicity

mα “1

CardpG´opαq X αG`

tpαq α´1q

of the common perpendicular α in pY, G˚q from pY´, G´˚ q to pY`, G`˚ q.Since the multiplicity mγ, γ1 is invariant under the diagonal action by left translations of

γ´10 γ´1 P Γ on pγ, γ1q, we may assume that γ “ γ0 “ id. Since the multiplicity mγ, γ1 is

invariant under right translation by γk`1´1, which stabilises D`, on the element γ1, we may

assume that γk`1 “ id. In particular, we have

opf1q “Ćope1q and tpfkq “ γ1 Ćtpekq .

200 19/12/2016

We use the basepoint x0 “ ope1q in the construction of the fundamental group and theBass-Serre tree of pY, G˚q, so that (see in particular [Bass, Eq. (1.3)])

V X “ž

βPπpY, G˚q : opβq“x0

β Gtpβq

andΓ “ π1pY, G˚q “ tβ P πpY, G˚q : opβq “ tpβq “ x0u .

Since an element in Γ which preserves D´ and γ1D` fixes pointwise its (unique) commonperpendicular in X, we have

ΓD´ X γ1ΓD`γ1´1

“ ΓD´ X Γγ1 D` “ pΓopf1q X ΓD´q X pΓtpfkq X Γγ1 D`q

“ pΓČope1q

X ΓD´q X pΓγ1 Ćtpekq

X Γγ1 D`q .

Note that ΓČope1q

X ΓD´ “ G´ope1q. By the construction of the edges in the Bass-Serre tree of a

graph of groups (see [Bass, page 11]), the vertex α Gtpekq is exactly the vertex tpfkq “ γ1 Ćtpekq.By [Bass, Eq. (1.4)], we hence have

αGtpekq α´1 “ Stabπ1pY,G˚qpαGtpekqq “ Γ

γ1 Ćtpekq.

Therefore mγ, γ1 “ mα.

(3) Let rαs “ Gopαq αGtpαq be a common perpendicular in pY, G˚q from pY´, G´˚ q topY`, G`˚ q, with representative α P πpY, G˚q, and let h0 e1 h1 . . . ek hk be a reduced wordwhose image in πpY, G˚q is α.

We define

‚ γ1 “ g e1 h´10 ,

‚ f1 “ γ´11 re1 ,

‚ assuming that γi and fi for some 1 ď i ď k ´ 1 are constructed, let

γi`1 “ g ei`1 h´1i g´1

ei γi and fi`1 “ γi`1´1

Ąei`1 ,

‚ with γk and fk constructed by induction, finally let γ1 “ γ´1k gek hk.

It is easy to check, using the equivalences in the proof of Lemma 12.16 (1) with γ “ γ0 “

γk`1 “ id, that the sequence pf1, . . . , fkq is the edge path of a common perpendicular in Xfrom D´ to γ1D` with origin Ćope1q and endpoint γ1 Ćtpekq.

If h0 is replaced by αh0 with α P G´ope1q, then by induction, f1, f2 . . . , fk are replaced byαf1, αf2, . . . , αfk and γ1 is replaced by αγ1. Note that pαf1, αf2, . . . , αfkq is then the commonperpendicular edge path from D´ “ αD´ to αγ1D`. If hk is replace by hk α with α P G`tpekq,then f1, f2 . . . , fk are unchanged, and γ1 is replaced by γ1 α. Note that γ1 αD` “ γ1D`.

Hence the map which associates to rαs the Γ-orbit of the common perpendicular in X fromD´ to γ1D` with edge path pf1, . . . , fkq is well defined. It is easy to see by construction thatthis map is the inverse of Θ. l

Theorem 12.15 now follows from Theorem 12.9. l

201 19/12/2016

12.6 Error terms for equidistribution and counting for metricand simplicial graphs of groups

In this Section, we give error terms to the equidistribution and counting results of Section12.4, given by Theorem 12.8 for metric trees (and their continuous time geodesic flows) and byTheorem 12.9 for simplicial trees (and their discrete time geodesic flows), under appropriatebounded geometry and rate of mixing properties.

Let pX, λq, X, Γ, rc, c, rFc, Fc, δc, D˘, D˘, λγ,γ1 , αγ,γ1 , α˘γ,γ1 , mγ,γ1 be as in Section 12.4.We first consider the simplicial case (when λ “ 1), for the discrete time geodesic flow.

Theorem 12.17. Let X be a locally finite simplicial tree without terminal vertices, let Γ bea nonelementary discrete subgroup of AutpXq, let rc be a system of conductances on X for Γand let D˘ be nonempty proper simplicial subtrees of X. Assume that the critical exponent δcis finite and positive, that the Gibbs measure mc (for the discrete time geodesic flow) is finiteand that the skinning measures σ˘D¯ are finite and nonzero. Assume furthermore that

(1) at least one of the following holds :

‚ ΓD˘zBD˘ is compact

‚ C ΛΓ is uniform and Γ is a lattice of C ΛΓ,

(2) there exists β P s0, 1s such that the discrete time geodesic flow on pΓzGX,mcq is expo-nentially mixing for the β-Hölder regularity.

Then there exists κ1 ą 0 such that for all ψ˘ P C βc pΓz

p

GXq, we have, as nÑ `8,

eδc ´ 1

eδcmc e

´δc nÿ

rγs PΓD´zΓΓD`0ădpD´,γ D`qďn

me, γ ecpαe, γq ψ´pΓα´e,γq ψ

`pΓα`γ´1,e

q

“

ż

ψ´ dσ`D´

ż

ψ` dσ´D` `O`

e´κ1 n ψ´β ψ

´β˘

and if ΓD˘zBD˘ is compact, then

ND´,D`pnq “eδc σ`D´ σ

´

D`

peδc ´ 1q mceδc n `O

`

epδc´κ1qn˘

.

Proof. We follow the scheme of proof of Theorem 12.7, replacing aspects of Riemannianmanifolds by aspects of simplicial trees as in the proof of Theorem 11.8. Let rψ˘ P C β

c p

p

GXq.In order to simplify the notation, let λγ “ λe,γ , αγ “ αe,γ , α´γ “ α´e,γ , α`γ “ α`

γ´1,eand

rσ˘ “ rσ˘D¯ .Let us first prove the following avatar of Equation (12.4), indicating only the required

changes in its proof: there exists κ0 ą 0 (independent of rψ˘) such that, as nÑ `8,

eδc ´ 1

eδcmc e

´δc nÿ

γPΓ, 0ăλγďn

ecpαγq rψ´pα´γ qrψ`pα`γ q

“

ż

B1`D´

rψ´ drσ`ż

B1´D`

rψ` drσ´ `Ope´κ0n rψ´β rψ`βq . (12.13)

202 19/12/2016

Most of the new work to be done in order to prove this formula concerns regularityproperties of the test functions that will be introduced later on. We start with the regularityof the fibration of the dynamical neighbourhoods.13

Let η,R ą 0 be such that η ă 1 ă lnR.

Lemma 12.18. Let Y be an R-tree and let D be a nonempty closed convex subset of Y .Then the restriction to V ˘

η,RpB1˘Dq of the fibration f˘D is (uniformly locally) Lipschitz, with

constants independent of η.

Proof. We assume for instance that ˘ “ `. Let `, `1 P V `η,RpB

1`Dq and let w “ f`D p`q, w

1 “

f`D p`1q.

Since the fiber over ρ P B1`D of the restriction to V `

η,RpB1`Dq of f

`D is V `ρ, η,R (see the end

of Section 2.5), there exist s, s1 P s ´ η, ηr such that gs` P B`pw,Rq and gs1

`1 P B`pw1, Rq, sothat gs`ptq “ wptq and gs

1

`1ptq “ w1ptq for all t ě lnR by the definition of the Hamenstädtballs. Up to permuting ` and `1, we assume that s1 ě s.

wď lnR

`1p0q

s1 ď η

`

`

w1

`p0q

`1

`1

By (the proof of) Lemma 10.11 (1), there exists a constant cR ą 0 depending only on Rsuch that if dp`, `1q ď cR and s2 “ dp`p0q, `1p0qq, then s2 “ s1 ´ s and the geodesic lines gs`and gs

1

`1 coincide at least on r´ lnR´ 1, lnR` 1s . In particular, we have

wplnRq “ `ps` lnRq “ `1ps1 ` lnRq “ w1plnRq .

Since the origin of w is the closest point on D to any point of wpr0,`8rq, we hence have thatwptq “ w1ptq for all t P r0, lnRs. Therefore (using Equation (2.5) for the last inequality),

dpw,w1q “

ż `8

lnRdpwptq, w1ptqq e´2 t dt “

ż `8

lnRdpgs`ptq, gs

1

`1ptqq e´2 t dt

“ e2s

ż `8

lnR`sdp`puq, gs

2

`1puqq e´2u du ď e2s dp`, gs2

`1q

ď e2s`

dp`, `1q ` dp`1, gs2

`1q˘

ď e2s pdp`, `1q ` s2q

“ e2s`

dp`, `1q ` dp`p0q, `1p0qq˘

,

so that the result follows from Lemma 10.11 (2).

Note that when Y is (the geometric realisation of) a simplicial tree, then we have s “ s1 “s2 “ 0 and the above computations simplify to give dpw,w1q ď dp`, `1q. l

13See Section 2.5 for notations.

203 19/12/2016

We fix R ą 0 big enough. With D˘ “ |D˘|1, we introduce the following modification ofthe test functions φ˘η :14

Φ˘η “ ph˘η,R

rψ˘q ˝ f¯D˘

1V ¯η,RpB1¯D˘q

.

As in Lemma 10.1, the functions Φ˘η are measurable and satisfy

ż

GXΦ˘η drmc “

ż

B1¯D˘

rψ˘ drσ¯ . (12.14)

Lemma 12.19. The maps Φ˘η are β-Hölder-continuous with

Φ˘η β “ Op rψ˘βq . (12.15)

Proof. Since X is a simplicial tree and η ă 1, we have V ˘w, η,R “ B˘pw,Rq for every w P B1˘D¯.

As seen above, there exists cR ą 0 depending only on R such that if ` P B˘pw,Rq and `1 P GXsatisfy dp`, `1q ď cR, then `1 P B˘pw,Rq. Hence (see Section 3.1) the characteristic function1V ˘η,RpB

1˘D¯q

is cR-locally constant, thus β-Hölder-continuous by Remark 3.2.By Assumption (1) in the statement of Theorem 12.17, the denominator of

h¯η,Rpwq “1

µW˘pwqpB˘pw,Rqq

is at least a positive constant depending only on R, hence h¯η,R is bounded by a constantdepending only on R. Since the map 1B˘pw,Rq is cR-locally constant, so is the map h¯η,R. Theresult then follows from Lemma 12.18 and Equation (3.1). l

In order to prove Equation (12.13), as in the proofs of Theorems 12.7 and 11.8, for allN P N, we estimate in two ways the quantity

IηpNq “Nÿ

n“0

eδc nÿ

γPΓ

ż

`PGXΦ´η pg

´tn2u`q Φ`η pgrn2sγ´1`q drmcp`q . (12.16)

On the one hand, as in order to obtain Equation (12.6), using now Assumption (2) in thestatement of Theorem 12.17 on the exponential mixing for the discrete time geodesic flow, ageometric sum argument and Equations (12.14) and (12.15), we have

IηpNq “eδcpN`1q

peδc ´ 1q mc

´

ż

B1`D´

rψ´ drσ`ż

B1´D`

rψ` drσ´ ` Ope´κN rψ´β rψ`βq

¯

. (12.17)

On the other hand, exchanging the summations over γ and n in the definition of IηpNq,we have

IηpNq “ÿ

γPΓ

Nÿ

n“0

eδc nż

GXΦ´η pg

´tn2u`q Φ`η pgrn2sγ´1`q drmcp`q .

14See Equation (10.1) for the definition of h˘η,R, that simplifies as h¯η,Rpwq “ pµW˘pwqpB˘pw,Rqqq´1 since

X is simplicial, as seen in Equation (11.24).

204 19/12/2016

With the simplifications in Step 3T of the proof of Theorem 11.1 given by the proof ofTheorem 11.8, if η ă 1

2 , if ` P GX belongs to the support of Φ´η ˝ g´tn2u Φ`η ˝ g

rn2s ˝ γ´1,setting w´ “ f`

D´p`q and w` “ f´

γD`p`q, we then have λγ “ n, w˘p0q “ α˘γ p0q and

w´ptn2uq “ w`p´rn2sq “ `p0q “ α´γ ptn2uq “ α`γ p´rn2sq ,

hencedpw˘, α˘γ q “ Ope´λγ q .

Therefore, since rψ˘ is β-Hölder-continuous,

| rψ˘pw˘q ´ rψ˘pv˘γ q | “ Ope´βλγ rψ˘βq .

Note that now Φ˘η “rψ˘ ˝ f¯

D˘φ˘η , so that

IηpNq “ÿ

γPΓ

`

rψ´pv´γ qrψ`pv`γ q `Ope´2βλγ rψ´β rψ

`βq˘

ˆ

Nÿ

n“0

eδc nż

GXφ´η pg

´tn2u`q φ`η pgrn2sγ´1`q drmcp`q .

Now if η ă 12 , Equation (12.13) with κ0 “ mint2β, κu follows as in Steps 3T and 4T of

the proof of Theorem 11.1 with the simplifications given by the proof of Theorem 11.8.The end of the proof of the equidistribution claim of Theorem 12.17 follows from Equation

(12.13) as the one of Theorem 12.7 from Equation (12.4).The counting claim follows from the equidistribution one by taking ψ˘ “ 1ΓVη,RpB1

¯D˘q,

which has compact support since ΓD˘zBD˘ is assumed to be compact, and is β-Hölder-continuous by previous arguments. l

Remarks. (i) Assume that rc “ 0, that the simplicial tree X1 with |X1|1 “ C ΛΓ is uniformwithout vertices of degree 2, that LΓ “ Z and that Γ is a geometrically finite lattice of X1. Thenall assumptions of Theorem 12.17 are satisfied by the results of Section 4.4 and by Corollary9.6. Therefore we have an exponentially small error term in the (joint) equidistribution of thecommon perpendiculars, and in their counting if ΓD˘zBD˘ is compact, see Example 12.10 (2).

(ii) Assume in this remark that Assumption (2) of the above theorem is replaced by theassumptions that C ΛΓ is uniform without vertices of degree 2, that LΓ “ 2Z, and that thereexists β P s0, 1s such that the square of the discrete time geodesic flow on pΓzGevenX,mcq isexponentially mixing for the β-Hölder regularity, for instance if Γ is geometrically finite byCorollary 9.6 (2). Then a similar proof (replacing the references to Theorem 11.8 by referencesto Theorem 11.10) shows that there exists κ1 ą 0 such that for all ψ˘ P C β

c pΓz

p

GXq, we have,as nÑ `8,

e2δc ´ 1

2 e2δcmc e

´δc nÿ

rγs PΓD´zΓΓD`0ădpD´,γ D`qďn

me, γ ecpαe, γq ψ´pΓα´e,γq ψ

`pΓα`γ´1,e

q

“

ż

ψ´ dσ`D´

ż

ψ` dσ´D` `O`

e´κ1 n ψ´β ψ

´β˘

205 19/12/2016

and if ΓD˘zBD˘ is compact, then

ND´,D`pnq “2 e2δc σ`D´ σ

´

D`

pe2δc ´ 1q mceδc n `O

`

epδc´κ1qn˘

.

Let us now consider the metric tree case, for the continuous time geodesic flow, where themain change is to assume a superpolynomial decay of correlations and hence get a superpoly-nomial error term.

Theorem 12.20. Let pX, λq, Γ, rc and D˘ be as in the beginning of this Section, and letD˘ “ |D˘|λ. Assume that the critical exponent δc is finite and positive, that the Gibbsmeasure mc (for the continuous time geodesic flow) is finite and that the skinning measuresσ˘D¯ are finite and nonzero. Assume furthermore that

(1) at least one of the following holds :

‚ ΓD˘zBD˘ is compact

‚ the metric subtree C ΛΓ is uniform and Γ is a lattice of C ΛΓ,

(2) there exists β P s0, 1s such that the continous time geodesic flow on pΓzGX,mcq hassuperpolynomial decay of β-Hölder correlations.

Then for every n P N there exists k P N such that for all ψ˘ P C k, βc pΓz

p

GXq, we have, asT Ñ `8,

δc mc e´δc T

ÿ

rγs PΓD´zΓΓD`0ădpD´,γ D`qďT

me, γ ecpαe, γq ψ´pΓα´e,γq ψ

`pΓα`γ´1,e

q

“

ż

Γz

p

GXψ´ dσ`D´

ż

Γz

p

GXψ` dσ´D` `O

`

T´n ψ´k, β ψ´k, β

˘

and if ΓD˘zBD˘ is compact, then for every n P N

ND´, D`pT q “σ`D´ σ

´

D`

δc mceδc T `O

`

eδc TT´n˘

.

Remark. Assume that rc “ 0, that the metric tree C ΛΓ is uniform, either that ΓzX is finiteand the length spectrum LΓ of Γ is 2-Diophantine or that Γ is a geometrically finite latticeof C ΛΓ and that LΓ is 4-Diophantine. Then all assumptions of Theorem 12.20 are satisfiedby the results of Section 4.4 and by Corollary 9.10. Therefore we have a superpolynomiallysmall error term in the (joint) equidistribution of the common perpendiculars (and in theircounting if ΓD˘zBD

˘ is compact).

Proof. The proof is similar to the one of Theorem 12.17, except that since the time is nowcontinuous, we need to regularise our test functions in the time direction in order to obtainthe regularity required for the application of the assumption on the mixing rate. We againuse the simplifying notation λγ “ λe,γ , αγ “ αe,γ , α´γ “ α´e,γ , α`γ “ α`

γ´1,eand rσ˘ “ rσ˘

D¯.

We fix n P N´t0u. Using the rapid mixing property, there exists a regularity k such thatfor all ψ,ψ1 P C k, β

b pΓzGXq we have as tÑ `8

covmc, t pψ,ψ1q “ Opt´N n ψk, β ψ

1k, βq , (12.18)

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where N P N´ t0u is a constant which will be made precise later on.Let us first prove that for all rψ˘ P C k, β

c p

p

GXq, we have, as T Ñ `8,

δc mc e´δc T

ÿ

γPΓ, 0ăλγďT

ecpαγq rψ´pα´γ qrψ`pα`γ q

“

ż

B1`D

´

rψ´ drσ`ż

B1´D

`

rψ` drσ´ `OpT´n rψ´k, β rψ`k, βq . (12.19)

In order to prove this formula, we introduce modified test functions, making them withbounded Hölder-continuous derivatives up to order k (by a standard construction) in the timedirection (the stable leaf and unstable leaf directions remain discrete). We fix R ą 0 bigenough.

For every η P s0, 1r , there exists a map x1η : R Ñ r0, 1s which has bounded β-Hölder-continuous derivatives up to order k, which is equal to 0 if t R r´η, ηs and to 1 if t Pr´η e´η, η e´ηs (when k “ 0, just take x1η to be continuous and linear on each remainingsegment r´η,´η e´ηs and rη e´η, ηs), such that, for some constant κ1 ą 0,

x1ηk, β “ Opη´κ1q .

Using leafwise this regularisation process, there exists χ˘η,R P C k, βb pGXq such that

‚ χ˘η,Rk, β “ Opη´κ1q,‚ 1V ¯

η e´η,RpB1¯D

˘qď χ˘η,R ď 1V ¯η,RpB

1¯D

˘q,

‚ for every w P B1¯D

˘, we haveż

V ¯w, η,R

χ˘η,R dν˘w “ ν˘w pV

¯w, η,Rq e

´Opηq “ ν˘w pV¯

w, η e´η , Rq eOpηq .

As in the proof of Theorem 12.7 in the manifold case, the new test functions are defined, with

H˘η,R : w P B1¯D

˘ ÞÑ1

ş

V ¯w, η,Rχ˘η,R dν

˘w,

byΦ˘η “ pH

˘η,R

rψ˘q ˝ f¯D˘

χ˘η,R : GX Ñ R .

Let pΦ˘η “ H˘η,R ˝ f¯

D˘χ˘η,R, so that Φ˘η “

rψ˘ ˝ f¯D˘

pΦ˘η . By the last two properties of theregularised maps χ˘η,R, we have, with φ¯η defined as in Equation (10.4),

φ˘η e´η

e´Opηq ď pΦ˘η ď φ˘η eOpηq . (12.20)

By Assumption (1), if R is large enough, by the definitions of the measures ν˘w , thedenominator of H˘η,Rpwq is at least c η where c ą 0. As in the proof of Theorem 12.7, thereexists κ2 ą 0 such that

ż

GXΦ¯η drmc “

ż

B1¯D

˘

rψ˘ drσ¯

andΦ˘η k, β “ Opη´κ

2

rψ˘k, βq .

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We again estimate in two ways as T Ñ `8 the quantity

IηpT q “

ż T

0eδc t

ÿ

γPΓ

ż

`PGXΦ´η pg

´t2`q Φ`η pgt2γ´1`q drmcp`q dt . (12.21)

Note that as T Ñ `8,

e´δc Tż T

1eδc t t´N n dt “ e´δc T

ż T 2

1eδc t t´N n dt` e´δc T

ż T

T 2eδc t t´N n dt

“ Ope´δc T 2q `OpT´N n`1q “ OpT´pN´1qnq .

Using Equation (12.18), an integration argument and the above two properties of the testfunctions, we hence have

IηpT q “eδc T

δc mc

´

ż

B1`D

´

rψ´ drσ`ż

B1´D

`

rψ` drσ´ ` OpT´pN´1qnη´2κ2 rψ´k, β rψ`k, βq

¯

.

(12.22)As in Step 3T of the proof of Theorem 11.1, for all γ P Γ and t ą 0 big enough, if ` P GX

belongs to the support of Φ´η ˝ g´t2 Φ`η ˝ g

t2 ˝ γ´1 (which is contained in the support ofφ´η ˝ g

´t2 φ`η ˝ gt2 ˝ γ´1 ), then we may define w´ “ f`

D´p`q and w` “ f´

γD`p`q.

By the property (iii) in Step 3T of the proof of Theorem 11.1, the generalised geodesiclines w´ and α´γ coincide, besides on s´8, 0s, at least on r0, t2´ηs, and similarly, w` and α`γcoincide, besides on r0,`8r , at least on r´ t

2 `η, 0s. Therefore, by an easy change of variableand since | t2 ´

λγ2 | ď η,

dpw´, α´γ q ď

ż `8

t2´ηdpw´psq, α´γ psqq e

´2s ds ď e´2p t2´ηq

ż `8

02s e´2s ds

“ Ope´tq “ Ope´λγ q .

Similarly, dpw`, α`γ q “ Ope´λγ q. Hence since rψ˘ is β-Hölder-continuous, we have

| rψ˘pw˘q ´ rψ˘pα˘γ q | “ Ope´βλγ rψ˘βq .

Therefore, as in the proof of Theorem 12.17, we have

IηpT q “ÿ

γPΓ

`

rψ´pα´γ qrψ`pα`γ q `Ope´2βλγ rψ´β rψ

`βq˘

ˆ

ż T

0eδ t

ż

`PGX

pΦ´η pg´t2`q pΦ`η pγ

´1gt2`q drmcp`q dt .

Finally, Equation (12.19) follows as in the end of the proof of Equation (12.4), usingEquations (12.20) and (11.16) instead of Equations (12.7) and (11.19), by taking η “ T´n

and N “ 2prκ2s` 1q.The end of the proof of the equidistribution claim of Theorem 12.20 follows from Equation

(12.19) as the one of Theorem 12.7 from Equation (12.4).The counting claim follows from the equidistribution one by taking ψ˘ to be β-Hölder-

continuous plateau functions around ΓVη,RpB1¯D˘q. l

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We are now in a position to prove one of the counting results in the introduction.

Proof of Theorem 1.9. Let X be the universal cover of Y, with fundamental group Γ for anindifferent choice of basepoint, and let D˘ be connected components of the preimages of Y˘in X. Assertion (1) of Theorem 1.9 follows from Theorem 12.20 and its subsequent Remark.Assertion (2) of Theorem 1.9 follows from Theorem 12.17 and its subsequent Remarks (ii)and (i), respectively if Y is bipartite or not. l

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Chapter 13

Geometric applications

In this final Chapter of Part II, we apply the equidistribution and counting results obtainedin the previous Chapters in order to study geometric equidistribution and counting problemsfor metric and simplicial trees concerning conjugacy classes in discrete isometry groups andclosed orbits of the geodesic flows.

13.1 Orbit counting in conjugacy classes for groups acting ontrees

In this Section, we study the orbital counting problem for groups acting on metric or simplicialtrees when we consider only the orbit points by elements in a given conjugacy class. We referto the Introduction for motivations and previously known results for manifolds (see [Hub1]and [PaP15]) and graphs (see [Dou] and [KeS]). The main tools we use are Theorem 12.8 forthe metric tree case and Theorem 12.17 for the simplicial tree case, as well as their error terms.We in particular obtain a much more general version of Theorem 1.12 in the Introduction.

Let pX, λq be a locally finite metric tree without terminal vertices, let X “ |X|λ be its geo-metric realisation, let x0 P V X and let Γ be a nonelementary discrete subgroup of AutpX, λq.1Let rc : EX Ñ R be a Γ-invariant system of conductances, let rFc and Fc be its associatedpotentials on T 1X and ΓzT 1X respectively, and let δc “ δΓ, F˘c

be its critical exponent.2

Let pµ˘x qxPX (respectively pµ˘x qxPV X) be Patterson densities for the pairs pΓ, F˘c q, and letrmc “ rmFc and mc “ mFc be the associated Gibbs measures on GX and ΓzGX (respectivelyGX and ΓzGX) for the continuous time geodesic flow (respectively the discrete time geodesicflow, when λ ” 1).3

Recall that the virtual centre ZvirtpΓq of Γ is the finite (normal) subgroup of Γ consistingof the elements γ P Γ acting by the identity on the limit set ΛΓ of Γ in B8X, see for instance[Cha, §5.1]. If ΛΓ “ B8X (for instance if Γ is a lattice), then ZvirtpΓq “ teu.

For any nontrivial element γ in Γ with translation length λpγq in X, let Cγ be

‚ the translation axis of γ if γ is loxodromic on X,

‚ the fixed point set of γ if γ is elliptic on X,

1See Section 2.7 for definitions and notation.2See Section 3.5 for definitions and notation.3See Sections 4.3 and 4.4 for definitions and notation.

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and let ΓCγ be the stabiliser of Cγ in Γ. In the simplicial case, Cγ is a simplicial subtree of X.Note that λpγq “ λpγ1γpγ1q´1q and γ1Cγ “ Cγ1γpγ1q´1 for all γ1 P Γ, and that for any x0 P X

dpx0, Cγq “dpx0, γx0q ´ λpγq

2. (13.1)

By the equivariance properties of the skinning measures, the total mass of the skinningmeasure4 σ´D where D “ pγ1Cγqγ1PΓΓCγ depends only on the conjugacy class K of γ in Γ, andwill be denoted by σ´K . This quantity, called the skinning measure of K, is positive unlessB8Cγ “ ΛΓ, which is equivalent to γ P ZvirtpΓq (and implies in particular that γ is elliptic).Furthermore, σ´K is finite if γ is loxodromic, and it is finite if γ is elliptic and ΓCγzpCγXC ΛΓqis compact. This last condition is in particular satisfied if Cγ X C ΛΓ itself is compact, andthis is the case for instance if, for some k ě 0, the action of Γ on X is k-acylindrical (see forinstance [Sel, GuL]), that is, if any element of Γ fixing a segment of length k in C ΛΓ is theidentity.

For every γ P Γ´ teu, we define

mγ “1

CardpΓx0 X ΓCγ q,

which is a natural multiplicity of γ, and equals 1 if the stabiliser of x0 in Γ is trivial (forinstance if Γ is torsion-free). Note that for every α P Γ, the real number mαγα´1 depends onlyon the double coset of α in Γx0zΓΓCγ .

The centraliser ZΓpγq of γ in Γ is contained in the stabiliser of Cγ in Γ. The index

iK “ rΓCγ : ZΓpγqs

depends only on the conjugacy class K of γ; it will be called the index of K. The index iK isfinite if γ is loxodromic (the stabiliser of its translation axis Cγ is then virtually cyclic), andalso finite if Cγ is compact (as for instance if the action of Γ on X is k-acylindrical for somek ě 0).

We define

cγ “kÿ

i“1

cpeiqλpeiq ,

where pe1, . . . , ekq is the shortest edge path from x0 to Cγ .

We finally define the orbital counting function in conjugacy classes, counting with multi-plicities and weights coming from the system of conductances, as

NK, x0ptq “ÿ

αPK, dpx0, αx0qďt

mα ecα .

for t P r0,`8r (simply t P N in the simplicial case). When the stabiliser of x0 in Γ is trivialand when the system of conductances c vanishes, we recover the definition of the Introduction.

Theorem 13.1. Let K be the conjugacy class of a nontrivial element γ0 of Γ, with finite indexiK, and with positive and finite skinning measure σ´K . Assume that δc is finite and positive.

4See Chapter 7 for definitions and notation.

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(1) Assume that mc is finite and mixing for the continuous time geodesic flow on ΓzX. Then,as tÑ `8,

NK, x0ptq „iK µ

`x0 σ´K e

´λpγ0q

2

δc mceδc2t .

If ΓCγzpCγ X C ΛΓq is compact when γ P K is elliptic and if there exists β P s0, 1s suchthat the continous time geodesic flow on pΓzGX,mcq has superpolynomial decay of β-Höldercorrelations, then the error term is O

`

t´n eδc2t˘

for every n P N.

(2) Assume that λ ” 1 and that mc is finite and mixing for the discrete time geodesic flow onΓzGX. Then, as nÑ `8,

NK, x0pnq „eδc iK µ

`x0 σ´K

peδc ´ 1q mceδct

n´λpγ0q2

u .

If ΓCγzpCγXC ΛΓq is compact when γ P K is elliptic and if there exists β P s0, 1s such that thediscrete time geodesic flow on pΓzGX,mcq is exponentially mixing for the β-Hölder regularity,then the error term is O

`

epδc´κqn2˘

for some κ ą 0.

One can also formulate a version of the above result for groups acting on bipartite simplicialtrees based on Theorem 12.12 and Remark (ii) following the proof of Theorem 12.17.

The error term in Assertion (1) holds for instance if rc “ 0, X is uniform, and either ΓzXis compact and the length spectrum LΓ is 2-Diophantine or Γ is a geometrically finite latticeof X whose length spectrum LΓ is 4-Diophantine, by the Remark following Theorem 12.20.When ΓzX is compact and Γ has no torsion (in particular, Γ has then a very restricted groupstructure, as it is then a free group), we thus recover a result of [KeS].

The error term in Assertion (2) holds for instance if rc “ 0, X is uniform with vertices ofdegrees at least 3, Γ is a geometrically finite lattice of X with length spectrum equal to Z, byRemark (i) following the proof of Theorem 12.17.

Theorem 1.12 in the introduction follows from this theorem, using Proposition 4.14 (3)and Proposition 4.15.

Proof. We only give a full proof of Assertion (1) of this theorem, Assertion (2) followssimilarly using Theorems 12.9 and 12.17 instead of Theorems 12.8 and 12.20.

The proof is similar to the proof of [PaP15, Theo. 8]. Let D´ “ tx0u and D` “ Cγ0 . LetD´ “ pγD´qγ1PΓΓD´ and D` “ pγD`qγPΓΓD` . By Equation (7.14), we have

σ`D´ “µ`x0

|Γx0 |.

By Equation (13.1), by the definition5 of the counting function ND´, D` and by the last claim

5See Section 12.4.

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of Theorem 12.8, we have, as tÑ `8,ÿ

αPK, 0ădpx0, αx0qďt

mα ecα “

ÿ

αPK, 0ădpx0, Cαqďt´λpγ0q

2

mα ecα

“ÿ

γPΓZΓpγ0q, 0ădpx0, γCγ0 qďt´λpγ0q

2

mγγ0γ´1 ecγγ0γ

´1

“ |Γx0 | iKÿ

γPΓx0zΓΓCγ0, 0ădpx0, γCγ0 qď

t´λpγ0q2

mγγ0γ´1 ecγγ0γ

´1

“ |Γx0 | iK ND´, D`` t´ λpγ0q

2

˘

„ |Γx0 | iKσ`D´ σ

´

D`

δc mceδc

t´λpγ0q2 .

Assertion (1) without the error term follows, and the error term statement follows similarlyfrom Theorem 12.20. l

Theorem 13.1 (1) without an explicit form of the multiplicative constant in the asymptoticis due to [KeS] under the strong restriction that Γ is a free group acting freely on X and ΓzXis a finite graph. The following result is due to [Dou, Thm. 1] in the very special case whenX is a regular tree and the group Γ has no torsion and finite quotient ΓzX.

Corollary 13.2. Let X be a regular simplicial tree with vertices of degree q ` 1 ě 3, letx0 P V X, let Γ be a lattice of X such that ΓzX is nonbipartite, and let K be the conjugacy classof a loxodromic element γ0 P Γ. Then, as nÑ `8,

ÿ

αPK, dpx0, αx0qďn

mα „λpγ0q

rZΓpγ0q : γZ0 s VolpΓzzXqqtn´λpγ0q

2u .

If we assume furthermore that Γ has no torsion, then the result holds also when ΓzX isbipartite. In this case, we have as nÑ `8,

Cardtα P K : dpx0, αx0q ď nu „λpγ0q

|ΓzV X|qtn´λpγ0q

2u .

Proof. Under these assumptions, taking c ” 0 in Theorem 13.1 so that the Gibbs measure isthe Bowen-Margulis measure, the discrete time geodesic flow on ΓzGX is finite and mixing byProposition 4.14 (3) and Proposition 4.15. We also have δc “ ln q. Using the normalisationof the Patterson density pµ˘x qxPV X to probability measures, Proposition 8.1 (3) and Equation(8.11), the result follows, since when γ is loxodromic,

VolpΓCγzzCγq “VolpγZzzCγq

rΓCγ : γZs“

λpγq

rΓCγ : ZΓpγqs rZΓpγq : γZs.

The claim for the bipartite pp` 1q-regular case follows from Remark 12.14. l

The value of C 1 given below Theorem 1.12 in the Introduction follows from this corollary.

We leave to the reader an extension with nonzero potential F of the results for manifoldsin [PaP15], along the lines of the above proofs.

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13.2 Equidistribution and counting of closed orbits on metricand simplicial graphs (of groups)

Classically, an important characterization of the Bowen-Margulis measure on compact nega-tively curved Riemannian manifolds is that it coincides with the weak-star limit of properlynormalised sums of Lebesgue measures supported on periodic orbits, see [Bowe1]. Under muchweaker assumptions than compactness, this result was extended to CATp´1q spaces with zeropotential in [Rob2] and to Gibbs measures in the manifold case in [PauPS, Thm. 9.11]. As acorollary of the simultaneous equidistribution results from Chapter 11, we prove in this Sec-tion the equidistribution towards the Gibbs measure of weighted closed orbits in quotients ofmetric and simplicial graphs of groups and as a corollary, in the standard manner, we obtainasymptotic counting results for weighted (primitive) closed orbits.

Let pX, λq be a locally finite metric tree without terminal vertices, and X “ |X|λ itsgeometric realisation. Let Γ be a nonelementary discrete subgroup of AutpX, λq. Let rc :EXÑ R be a Γ-invariant system of conductances, and c : ΓzEXÑ R its induced function.

Given a periodic orbit g of the geodesic flow on ΓzGX, if pe1, . . . , ekq is the sequence ofedges followed by g, we denote by Lg the Lebesgue measure along g, by λpgq the length of gand by cpgq its period for the system of conductances c:

λpgq “kÿ

i“1

λpeiq and cpgq “kÿ

i“1

λpeiq cpeiq .

Let Perptq be the set of periodic orbits of the geodesic flow on ΓzGX and let Per1ptq be thesubset of prime periodic orbits.

Theorem 13.3. Assume that the critical exponent δc of c is finite and positive and that theGibbs measure mc of c is finite and mixing for the continuous time geodesic flow. As tÑ `8,the measures

δc e´δc t

ÿ

gPPer1ptq

ecpgqLg

andδc t e

´δc tÿ

gPPer1ptq

ecpgqLg

`pgq

converge to mcmc

for the weak-star convergence of measures. If Γ is geometrically finite, theconvergence holds for the narrow convergence.

We conjecture that if Γ is geometrically finite and if its length spectrum is 4-Diophantine6,then for all n P N and β P s0, 1s, there exists k P N and an error term of the form Optn ψk, βq

for these equidistribution claims evaluated on any ψ P C b, βc pΓzGXq. But since we will not

need this result and since the proof is likely to be very long, we do not address the problemhere.

Proof. Let rFc and Fc be the potentials on T 1X and ΓzT 1X respectively associated with7 c,and note that the period8 of a periodic orbit g for the geodesic flow on ΓzGX satisfies

cpgq “ LgpF7c q “ PerFcpγq ,

6See definition in Section 9.3.7See Section 3.5.8See Section 3.2.

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where F 7c is the composition of the canonical map ΓzGX Ñ ΓzT 1X with Fc : ΓzT 1X Ñ R,and γ P Γ is the loxodromic element of Γ whose conjugacy class corresponds to g.

Let HΓ,t be the subset of Γ that consists of loxodromic elements whose translation lengthis at most t, and let H 1

Γ,t be the subset of HΓ,t that consists of the primitive elements. Thefirst claim is equivalent to the following assertion: we have

δc e´δc t

ÿ

γPH 1Γ,t

ePerFc pγqLg˚á

mc

mc(13.2)

as t Ñ `8. We proceed with the proof of the convergence claimed in Equation (13.2) as in[PauPS, Thm. 9.11]. We first prove that

ν2t “ δc e´δc t

ÿ

γPHΓ,t

ePerFc pγqLg˚á

mc

mc. (13.3)

We then refer to Step 2 of the proof of [PauPS, Thm. 9.11] for the fact that the contributionof the periods that are not primitive is negligible. Although the proof in loc. cit. is writtenfor manifolds, the arguments are directly applicable for any CATp´1q space X and potentialF satisfying the HC-property.9 In particular, the use of Proposition 5.13 (i) and (ii) of loc.cit. in the proof of Step 2 in loc. cit. is replaced now by the use of Theorem 4.5 (1) and (4)respectively.

Let us fix x P X. Let

V pxq “

pξ, ηq P pX Y B8Xq2 : ξ ‰ η, x P sξ, ηr

(

,

which is an open subset of X Y B8X. Note that the family pV pyqqyPX covers the set of pairsof distinct points of B8X. For every t ą 0, let νt be the measure on pX Y B8Xq2 defined by

νt “ δc mc e´δct

ÿ

γPΓ : dpx,γxqďt

eşγxx

rFc∆γ´1x b∆γx .

The measures νt weak-star converge to µ´x b µ`x as t Ñ `8 by Corollary 11.2 (taking in itsstatement y “ x).

Let γ˘ be the attracting and repelling fixed points of any loxodromic element γ P Γ. Let

ν3t “ δc mc e´δc t

ÿ

γPHΓ, t

ePerFc pγq∆γ´ b∆γ` .

Since X is an R-tree, every element γ P Γ such that x P sγ´1x, γxr is loxodromic, and we have

dpx, γxq “ λpγq and

ż γx

x

rFc “ PerFcpγq .

If furthermore dpx, γxq is big, then γ´1x and γx are respectively close to γ´ and γ` inX Y B8X.

Hence, for every continuous map ψ : pX Y B8Xq2 Ñ r0,`8r with support contained in

V pxq, and for every ε ą 0, if t is big enough, we have

e´ενtpψq ď ν3t ψq ď eενtpψq .

9See Definition 3.4.

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Using Hopf’s parametrisation with basepoint x, we have ν3t b ds “ mc ν2t , and the support

of any continuous function with compact support on GX may be covered by finitely manyopen sets V pxq ˆ R where x P X. The proof of the claim (13.3) now follows as in Step 1 ofthe proof of Theorem [PauPS, Thm. 9.11].

The second claim of Theorem 13.3 follows from the first one in the same way as in [PauPS,Thm. 9.11] to which we refer for the proof. l

In a similar way, replacing in the above proof Corollary 11.2 of Theorem 11.1 by the similarcorollary of Theorem 11.8 with D´ “ pγxqγPΓ and D´ “ pγyqγPΓ for any x, y P V X, we getthe following analogous result for simplicial trees.

Theorem 13.4. Let X be a locally finite simplicial tree without terminal vertices, let Γ bea nonelementary discrete subgroup of AutpXq and let rc : EX Ñ R be a Γ-invariant systemof conductances. Assume that the critical exponent δc of c is finite and positive and that theGibbs measure mc is finite and mixing for the discrete time geodesic flow. As n Ñ `8, themeasures

eδc ´ 1

eδce´δcn

ÿ

gPPer1pnq

ecpgqLg

andeδc ´ 1

eδcn e´δcn

ÿ

gPPer1pnq

ecpgqLg

`pgq

converge to mcmc

for the weak-star convergence of measures. If Γ is geometrically finite, theconvergence holds for narrow convergence. l

In the special case when ΓzX is a compact graph and F “ 0, the following immediatecorollary of Theorem 13.3 is proved in [Gui], and it follows from the results of [ParP].10 Thereare also some works on non-backtracking random walks with related results. For example,for regular finite graphs, [LuPS1] and [Fri1] (see [Fri2, Lem. 2.3]) give an expression of theirreducible trace which is the number of closed walks of a given length.

Corollary 13.5. Let pX, λq be a locally finite metric tree without terminal vertices. Let Γ bea geometrically finite discrete subgroup of AutpX, λq. Let c : EXÑ R be a Γ-invariant systemof conductances, with finite and positive critical exponent δc.(1) If the Gibbs measure mc is finite and mixing for the continuous time geodesic flow, then

ÿ

gPPer1ptq

ecpgq „eδc t

δc t

as tÑ `8.(2) If λ “ 1 and if the Gibbs measure mc is finite and mixing for the discrete time geodesicflow, then

ÿ

gPPer1pnq

ecpgq „eδc

eδc ´ 1

eδc n

n

as nÑ `8. l

10See the introduction of [Sha] for comments.

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Part III

Arithmetic applications

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Chapter 14

Fields with discrete valuations

Let pK be a non-Archimedean local field. Basic examples of such fields are the field of formalLaurent series over a finite field, and the field of p-adic numbers (see Examples 14.1 and14.2). In Part III of this book, we apply the geometric equidistribution and counting resultsfor simplicial trees given in Part II, in order to prove arithmetic equidistribution and countingresults in such fields pK. The link between the geometry and the algebra is provided by theBruhat-Tits tree of pPGL2, pKq, see Chapter 15.1. We will only use the system of conductancesequal to 0 in this Part III.

In the present Chapter, before embarking on our arithmetic applications, we recall basicfacts on local fields for the convenience of the geometer reader. For more details, we refer forinstance to [Ser2, Gos]. We refer to [BrPP] for an announcement of the results of Part III,with a presentation different from the one in the Introduction.

We will only give results for the algebraic group G “ PGL2 over pK and special discretesubgroups Γ of PGL2p pKq, even though the same methods give equidistribution and countingresults when G is any semisimple connected linear algebraic group over pK of pK-rank 1 and Γany lattice in G “ Gp pKq.

14.1 Local fields and valuations

Let F be a field and let Fˆ “ pF ´ t0u,ˆq be its multiplicative group. A surjective groupmorphism v : Fˆ Ñ Z to the additive group Z, that satisfies

vpa` bq ě mintvpaq, vpbqu

for all a, b P Fˆ, is a (normalised discrete) valuation v on F . We make the usual conventionand extend the definition of v to F by setting vp0q “ `8. Note that vpa`bq “ mintvpaq, vpbquif vpaq ‰ vpbq. When F is an extension of a finite field k, the valuation v vanishes on kˆ.

The subringOv “ tx P F : vpxq ě 0u

is the valuation ring (or local ring) of F or of v.The maximal ideal

mv “ tx P F : vpxq ą 0u

of Ov is principal and it is generated as an ideal of Ov by any element πv P F with

vpπvq “ 1

221 19/12/2016

which is called a uniformiser of F .The residual field of the valuation v is

kv “ Ovmv .

When kv is finite, the valuation v defines a (normalised, non-Archimedean) absolute value | ¨ |von F by

|x|v “ |kv|´vpxq ,

with the convention that |kv|´8 “ 0. This absolute value induces an ultrametric distance onF by

px, yq ÞÑ |x´ y|v .

Let Fv be the completion of F with respect to this distance. The valuation v of F uniquelyextends to a (normalised discrete) valuation on Fv, again denoted by v.

Example 14.1. Let K “ FqpY q be the field of rational functions in one variable Y withcoefficients in a finite field Fq of order a positive power q of a positive prime p in Z, let FqrY sbe the ring of polynomials in one variable Y with coefficients in Fq, and let v8 : Kˆ Ñ Z bethe valuation at infinity of K, defined on every P Q P K with P P FqrY s and Q P FqrY s´t0uby

v8pP Qq “ degQ´ degP .

The absolute value associated with v8 is

|P Q|8 “ qdegP´degQ .

The completion of K for v8 is the field Kv8 “ FqppY ´1qq of formal Laurent series in onevariable Y ´1 with coefficients in Fq. The elements x in FqppY ´1qq are of the form

x “ÿ

iPZxi Y

´i

where xi “ 0 P Fq for i P Z small enough. The valuation at infinity of FqppY ´1qq extendingthe valuation at infinity of FqpY q is

v8pxq “ supti P Z : @ j ă i, xj “ 0u ,

that is,

v8p8ÿ

i“i0

xiY´iq “ i0

if xi0 ‰ 0. The valuation ring of v8 is the ring Ov8 “ FqrrY ´1ss of formal power series inone variable Y ´1 with coefficients in Fq. The element πv8 “ Y ´1 is a uniformizer of v8, theresidual field Ov8πv8Ov8 of v8 is kv8 “ Fq.

Example 14.2. Given a positive prime p P Z, the field of p-adic numbers Qp is the completionof Q with respect to the absolute value | ¨ |p of the p-adic valuation vp defined by setting

vpppna

bq “ n

when n P Z, a, b P Z´ t0u are not divisible by p. Then the valuation ring Ovp “ Zp of Qp isthe closure of Z for the absolute value | ¨ |p, πvp “ p is a uniformiser, and the residual field iskvp “ ZppZp “ Fp, a finite field of order p.

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A field endowed with a valuation is a non-Archimedean local field if it is complete withrespect to its absolute value and if its residual field is finite.1 Its valuation ring is then acompact open additive subgroup. Any non-Archimedean local field is isomorphic to a finiteextension of the p-adic field Qp for some prime p, or to an extension of Fp with transcendencedegree 1 for some prime p.

The basic case of extensions of Fp with transcendence degree 1 is described in Example14.1 above, and the general case of the discussed in Section 14.2 below. The geometer readermay skip Section 14.2 and use only Example 14.1 in the remainder of Part III (using g “ 0when the constant g occurs).

14.2 Global function fields

In this Section, we fix a finite field Fq with q elements, where q is a positive power of a positiveprime p P Z, and we recall the definitions and basic properties of a function field K over Fq,its genus g, its valuations v, its completion Kv for the associated absolute value | ¨ |v and theassociated affine function ring Rv. See for instance [Gos, Ros] for the content of this Section.

Let K be a (global) function field over Fq, which can be defined in two equivalent ways as

(1) the field of rational functions on a geometrically irreducible smooth projective curve Cover Fq, or

(2) an extension of Fq of transcendence degree 1, in which Fq is algebraically closed.

There is a bijection between the set of closed points of C and the set of (normalised discrete)valuations of its function field K, the valuation of a given element f P K being the order ofthe zero or the opposite of the order of the pole of f at the given closed point. We fix suchan element v from now on. We denote by g the genus of the curve C.

In the basic Example 14.1, C is the projective line P1 over Fq which is a curve of genusg “ 0, and the closed point associated with the valuation at infinity is the point at infinityr1 : 0s.

We denote by Kv the completion of K for v, and by

Ov “ tx P Kv : vpxq ě 0u

the valuation ring of (the unique extension to Kv) of v. We choose a uniformizer πv of v. Wedenote by kv “ OvπvOv the residual field of v, which is a finite field of order

qv “ |kv| .

The field kv is from now on identified with a fixed lift in Ov (see for instance [Col, Théo. 1.3]),and is an extension of the field of constants Fq. The degree of this extension is denoted bydeg v, so that

qv “ qdeg v .

We denote by | ¨ |v the (normalised) absolute value associated with v : for every x P Kv, wehave

|x|v “ pqvq´vpxq “ q´vpxq deg v .

1There are also two Archimedean local fields C and R, see for example [Cas].

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Every element x P Kv is2 a (converging) Laurent series x “ř

iPZ xi pπvqi in the variable πv

over kv, where xi P kv is zero for i P Z small enough. We then have

|x|v “ pqvq´ suptjPZ : @ iăj, xi“0u , (14.1)

and Ov consists of the (converging) power series x “ř

iPN xi pπvqi (where xi P kv) in the

variable πv over kv.

We denote by Rv the affine algebra of the affine curve C´ tvu, consisting of the elementsof K whose only poles are at the closed point v of C. Its field of fractions is equal to K, hencewe will often write elements of K as xy with x, y P Rv and y ‰ 0. In the basic Example 14.1,we have Rv8 “ FqrY s. Note that

Rv X Ov “ Fq , (14.2)

since the only rational functions on C whose only poles are at v and whose valuation at v isnonnegative are the constant ones. We have (see for instance [Ser3, II.2 Notation], [Gos, page63])

pRvqˆ “ pFqqˆ . (14.3)

The following result is immediate when C “ P1, since then Rv ` Ov “ Kv.

Lemma 14.3. The dimension of the quotient vector space KvpRv ` Ovq over Fq is equal tothe genus g of C.

Proof. (J.-B. Bost) We refer for instance to [Ser1] for background on sheaf cohomology. Wedenote in the same way the valuation v and the corresponding closed point on C.

Let O “ K X Ov be the discrete valuation ring of v restricted to K. Since K is dense inKv and Ov is open and contains 0, we have Kv “ K ` Ov. Therefore the canonical map

KpRv ` Oq Ñ KvpRv ` Ovq

is a linear isomorphism over Fq. Let us hence prove that dimFq KpRv ` Oq “ g.In what follows, V ranges over the affine Zariski-open neighbourhoods of v in C, ordered

by inclusion. Let OC be the structural sheaf of C. Note that by the definition of Rv, since thezeros of elements of Kˆ are isolated and by the relation between valuations of K and closedpoints of C,

Rv “ H0pC´ tvu,OCq, K “ limÝÑV

H0pV ´ tvu,OCq and O “ limÝÑV

H0pV ,OCq .

Since V and C´ tvu are affine curves, we have H1pC´ tvu,OCq “ H1pV ,OCq “ 0. By theMayer-Vietoris exact sequence since tC´tvu,V u covers C, we hence have an exact sequence

H0pC,OCq Ñ H0pC´ tvu,OCq ˆH0pV ,OCq Ñ H0pV ´ tvu,OCq Ñ H1pC,OCq .

Therefore

KpRv ` Oq “ limÝÑV

H0pV ´ tvu,OCq`

H0pC´ tvu,OCq `H0pV ,OCq

˘

» H1pC,OCq .

2See for instance [Col, Coro. 1.6]

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Since dimFq H1pC,OCq “ g by one definition of the genus of C, the result follows. l

Recall that Rv is a Dedekind ring.3 In particular, every nonzero ideal (respectively frac-tional ideal) I of Rv may be written uniquely as I “

ś

p pvppIq where p ranges over the prime

ideals in Rv and vppIq P N (respectively vppIq P Z), with only finitely many of them nonzero.By convention I “ Rv if vppIq “ 0 for all p. For every x, y P Rv (respectively x, y P K), wedenote by

xx, yy “ xRv ` y Rv

the ideal (respectively fractional ideal) of Rv generated by x, y. If I, J are nonzero fractionalideals of Rv, we have

I X J “ź

p

pmaxtvppIq, vppJqu and I ` J “ź

p

pmintvppIq, vppJqu .

We define the (absolute) norm of a nonzero ideal I “ś

p pvppIq of Rv by

NpIq “ rRv : Is “ |RvI| “ź

p

qvppIqdeg p ,

where deg p is the degree of the field Rvp over Fq, so that NpRvq “ 1. By conventionNp0q “ 0. This norm is multiplicative:

NpIJq “ NpIqNpJq ,

and the norm of a nonzero fractional ideal I “ś

p pvppIq of Rv is defined by the same formula.

Note that if paq is the principal ideal in Rv generated by a, we define Npaq “ N`

paq˘

. Wehave (see for instance [Gos, page 63])

Npaq “ |a|v . (14.4)

Dedekind’s zeta function of K is (see for instance [Gos, §7.8] or [Ros, §5])

ζKpsq “ÿ

I

1

NpIqs

if Re s ą 1, where the summation is over the nonzero ideals I of Rv. By for instance [Ros, §5],it has an analytic continuation on C´ t0, 1u with simple poles at s “ 0, s “ 1. It is actuallya rational function of q´s. In particular, if K “ FqpY q, then (see [Ros, Theo. 5.9])

ζFqpY qp´1q “1

pq ´ 1qpq2 ´ 1q. (14.5)

We denote by HaarKv the Haar measure of the (abelian) locally compact topologicalgroup pKv,`q, normalised so that HaarKvpOνq “ 1.4 The Haar measure scales as followsunder multiplication: for all λ, x P Kv, we have

dHaarKvpλxq “ |λ|v dHaarKvpxq . (14.6)

Note that any fractional ideal I of Rv is a discrete subgroup of pKv,`q, and we will againdenote by HaarKv the Haar measure on the compact group KvI which is induced by theabove normalised Haar measure of Kv.

3See for instance [Ser3, II.2 Notation]. We refer for instance to [Nar, §1.1] for background on Dedekindrings.

4Other normalisations are useful when considering Fourier transforms, see for instance Tate’s thesis [Tat].

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Lemma 14.4. For every fractional ideal I of Rv, we have

HaarKvpKvIq “ q g´1 NpIq .

Proof. By the scaling properties of the Haar measure, we may assume that I is an idealin Rv. By Lemma 14.3, we have Card KvpRv ` Ovq “ qg. By Equation (14.2) and by thenormalisation of the Haar measure, we have

HaarKvpRv ` OvqRv “ HaarKv OvpRv X Ovq “ HaarKv OvFq “1

q.

HenceHaarKvpKvRvq “ Card

`

KvpRv ` Ovq˘

HaarKvpRv ` OvqRv “ qg´1 .

Since HaarKvpKvIq “ NpIq HaarKvpKvRvq, the result follows. l

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Chapter 15

Bruhat-Tits trees and modular groups

In this Chapter, we give the background information and preliminary results on the mainlink between the geometry and the algebra used for our arithmetic applications: the (discretetime) geodesic flow on quotients of Bruhat-Tits trees by arithmetic lattices.

We denote the image in PGL2 of an elementˆ

a bc d

˙

P GL2 by„

a bc d

P PGL2.

15.1 Bruhat-Tits trees

Let Kv be a non-Archimedean local field, with valuation v, valuation ring Ov, choice ofuniformiser πv, and residual field kv of order qv (see Section 14.1 for definitions).

In this Section, we recall the construction and basic properties of the Bruhat-Tits tree Xvof pPGL2,Kvq, see for instance [Tit]. We use its description given in [Ser3], to which we referfor proofs and further information.

An Ov-lattice Λ in the Kv-vector space KvˆKv is a rank 2 free Ov-submodule of KvˆKv,generating Kv ˆKv as a vector space. The Bruhat-Tits tree Xv of pPGL2,Kvq is the graphwhose set of vertices V Xv is the set of homothety classes (under pKvq

ˆ) rΛs of Ov-lattices Λin KvˆKv, and whose non-oriented edges are the pairs tx, x1u of vertices such that there existrepresentatives Λ of x and Λ1 of x1 for which Λ Ă Λ1 and Λ1Λ is isomorphic to OvπvOv. IfK is any field endowed with a valuation v whose completion is Kv, then the similarly definedBruhat-Tits tree of pPGL2,Kq coincides with Xv, see [Ser3, p. 71].

The graph Xv is a regular tree of degree |P1pkvq| “ qv ` 1. In particular, the Bruhat-Titstree of pPGL2,Qpq is regular of degree p ` 1, and if Kv “ FqppY ´1qq and v “ v8, then theBruhat-Tits tree Xv of pPGL2,Kvq is regular of degree q ` 1. More generally, if Kv is thecompletion of a function field over Fq endowed with a valuation v as in Section 14.2, then theBruhat-Tits tree of pPGL2,Kvq is regular of degree qv ` 1.

The standard base point ˚v of X is the homothety class rOvˆOvs of the Ov-lattice OvˆOv,generated by the canonical basis of Kv ˆKv. In particular, we have

dp˚v, rOv ˆ xOvsq “ |vpxq| (15.1)

for every x P pKvqˆ. The link

lkp˚vq “ ty P V Xv : dpy, ˚vq “ 1u

of ˚v in Xv identifies with the projective line P1pkvq.

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The left linear action of GL2pKvq onKvˆKv induces a faithful, vertex-transitive left actionby automorphisms of PGL2pKvq on Xv. The stabiliser in PGL2pKvq of ˚v is PGL2pOvq. Wewill hence identify PGL2pKvqPGL2pOvq with V Xv by the map g PGL2pOvq ÞÑ g ˚v.

We identify the projective line P1pKvq with Kv Y t8u using the map Kvpx, yq ÞÑxy , so

that8 “ r1 : 0s .

The projective action of GL2pKvq or PGL2pKvq on P1pKvq is the action by homographies1

on Kv Y t8u, given by pg, zq ÞÑ g ¨ z “ a z`bc z`d if g “

ˆ

a bc d

˙

P GL2pKvq, or g “„

a bc d

P

PGL2pKvq. As usual we define 8 ÞÑ ac and ´d

c ÞÑ 8.There exists a unique homeomorphism between the boundary at infinity B8Xv of Xv and

P1pKvq such that the (continuous) extension to B8Xv of the isometric action of PGL2pKvq onXv corresponds to the projective action of PGL2pKvq on P1pKvq. From now on, we identifyB8Xv and P1pKvq by this homeomorphism. Under this identification, Ov consists of thepositive endpoints `` of the geodesic lines ` of Xv with negative endpoint `´ “ 8 that passthrough the vertex ˚v (see the picture below).

8

B8Xv

0

Ov

0 0˚v

x0

x2

0

x1

xi0

x P Kv

H8

Let H8 be the horoball centred at 8 P B8Xv whose associated horosphere passes through˚v. There is a unique labeling of the edges of Xv by elements of P1pkvq “ kv Y t8u such that• the label of any edge of Xv pointing towards 8 P B8Xv is 8,• for any x “

ř

iPZ xi pπvqi P Kv, the sequence pxiqiPZ is the sequence of the labels of the

(directed) edges that make up the geodesic line s8, xr oriented from 8 towards x• x0 is the label of the edge of s8, xr exiting the horoball H8.We refer to [Pau1, Sect. 5] when Kv “ FqppY ´1qq and v “ v8.

For all η, η1 P Kv “ B8Xv ´ t8u, we have

|η ´ η1|v “ dH8pη, η1qln qv (15.2)

by the definitions of the absolute value | ¨ |v and of Hamenstädt’s distance, see Equation(14.1) and the above geometric interpretation, and Equation (2.8). Note that in [Pau1],Hamenstädt’s distance in a regular tree is defined in a different way: In that reference, thedistance |η ´ η1|v equals Hamenstädt’s distance between η and η1.

1or linear fractional transformations

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In particular, the Hölder norms ψβ, | ¨ ´ ¨ |v and ψβ1, dH8of a function ψ : Kv Ñ R,

respectively for the distance px, yq ÞÑ |x ´ y|v and dH8on Kv, are related by the following

formula:@ β P s0,

1

ln qvs, ψβ, | ¨ ´ ¨ |v “ ψβ ln qv , dH8

. (15.3)

The group PGL2pKvq acts simply transitively on the set of ordered triples of distinctpoints in B8Xv “ P1pKvq. In particular, it acts transitively on the space GXv of discretegeodesic lines in Xv. The stabiliser under this action of the geodesic line (from 8 “ r1 : 0s to0 “ r0 : 1s)

`˚ : n ÞÑ rOv ˆ pπvq´nOvs

is the maximal compact-open subgroup

ApOvq “

!

„

a 00 d

: a, d P pOvqˆ)

of the diagonal group

ApKvq “

!

„

a 00 d

: a, d P pKvqˆ)

.

We will hence identify PGL2pKvqApOvq with GXv by the mapping rΞ : gApOvq ÞÑ g `˚. Define

av “

„

1 00 π´1

v

,

which belongs to ApKvq and centralises ApOvq. The homeomorphism rΞ is equivariant for theactions on the left of PGL2pKvq on PGL2pKvqApOvq and GXv. It is also equivariant for theactions on PGL2pKvqApOvq under translations on the right by pavqZ and on GXv under thediscrete geodesic flow pgnqnPZ: for all n P Z and x P PGL2pKvqApOvq, we have

rΞ px avnq “ gn rΞ pxq . (15.4)

Furthermore, the stabiliser in PGL2pKvq of the ordered pair of endpoints p`˚´ “ 8, `˚` “ 0qof `˚ in B8Xv “ P1pKvq is ApKvq. Therefore any element γ P PGL2pKvq which is loxodromicon Xv is diagonalisable over Kv. By [Ser3, page 108], the translation length on Xv of γ0 “„

a 00 d

is

λpγ0q “ |vpaq ´ vpdq| . (15.5)

Note that if rγ0 “

ˆ

a 00 d

˙

P GL2pKvq is a representative of γ0 such that det rγ0 P pOvqˆ,

then as 0 “ vpdet rγ0q “ vpadq “ vpaq ` vpdq and since vpaq ‰ vpdq if λpγ0q ‰ 0, we havevptr rγ0q “ vpa` dq “ maxtvpaq, vpdqu and vpaq ´ vpdq “ 2vpaq “ ´2vpdq. Thus,

λpγ0q “ 2|vptr rγ0q| . (15.6)

By conjugation, this formula is valid if γ0 P PGL2pKvq is loxodromic on Xv and representedby rγ0 P GL2pKvq such that det rγ0 P pOvq

ˆ.

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Let H be a horoball in Xv whose boundary is contained in V Xv and whose point atinfinity ξ is different from 8. The height of H is

ht8pH q “ maxtβ8px, ˚vq : x P BH u P Z ,

which is the signed distance between H8 and H .2 It is attained at the intersection pointwith BH of the geodesic line from 8 to ξ, which is then called the highest point of H . Notethat the height of H is invariant under the action of the stabiliser of H8 in PGL2pKvq onthe set of such horoballs H .

The following lemma is a generalisation of [Pau1, Prop. 6.1] that covers the particular caseof K “ FqpY q and v “ v8.

Lemma 15.1. Assume that Kv is the completion of a function field K over Fq endowed with

a valuation v, with associated affine function ring Rv. For every γ “„

a bc d

P PGL2pKq

with a, b, c, d P K such that ad´ bc P pOvqˆ and c ‰ 0, the image of H8 by γ is the horoball

centred at ac P K Ă Kv “ B8Xv ´ t8u with height

ht8pγH8q “ ´2 vpcq .

Proof. It is immediate that γ8 “ ac under the projective action. Up to multiplying γ on

the left by„

1 ´ac

0 1

P PGL2pKq, which does not change c nor the height of γH8, we may

assume that a “ 0 and that b has the form c´1u with u “ bc ´ ad P pOvqˆ. Multiplying γ

on the right by„

1 ´dc

0 1

P PGL2pKq preserves γH8 and does not change a “ 0, b “ c´1u or

c. Hence we may assume that d “ 0. Since γ then exchanges the points 8 and 0 in B8Xv,the highest point of γH8 is γ˚v. Assuming first that 0, γ˚v, ˚v,8 are in this order on thegeodesic line from 0 to 8, we have by Equation (15.1)

ht8pγH8q “ dp˚v, γ˚vq “ dprOv ˆ Ovs, rc´1uOv ˆ cOvsq

“ dprOv ˆ Ovs, rOv ˆ c2Ovsq “ ´vpc

2q “ ´2 vpcq .

If 0, γ˚v, ˚v,8 are in the opposite order, then the same computation holds, up to replacingthe distance d by its opposite ´d. l

15.2 Modular graphs of groups

Let K be a function field over Fq, let v be a (normalised discrete) valuation of K, let Kv bethe completion of K associated with v, and let Rv be the affine function ring associated withv (see Section 14.2 for definitions).

The group Γv “ PGL2pRvq is a lattice in the locally compact group PGL2pKvq, and alattice3 of the Bruhat-Tits tree Xv of pPGL2,Kvq, called the modular group at v of K. Thequotient graph ΓvzXv is called the modular graph at v of K, and the quotient graph of groups4

ΓvzzXv is called the modular graph of groups at v of K. We refer to [Ser3] for background2See the definition of signed distance just above Lemma 11.12.3See Section 2.7 for a definition.4See Section 2.7 for a definition.

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information on these objects, and for instance to [Pau1] for a geometric treatment whenK “ FqpY q and v “ v8.

By for instance [Ser3], the set of cusps ΓvzP1pKq is finite, and ΓvzXv is the disjoint union ofa finite connected subgraph containing Γv˚v and of maximal open geodesic rays hzp s0,`8rq,for z “ Γvrz P ΓvzP1pKq, where hz (called a cuspidal ray) is the image by the canonicalprojection Xv Ñ ΓvzXv of a geodesic ray whose point at infinity in P1pKq Ă B8Xv is equalto rz. Conversely, any geodesic ray whose point at infinity lies in P1pKq Ă B8Xv contains asubray that maps injectively by the canonical projection Xv Ñ ΓvzXv.

The group Γv “ PGL2pRvq is a geometrically finite lattice by for instance [Pau2].5 Theset of bounded parabolic fixed points of Γv is exactly P1pKq Ă B8Xv, and the set of conicallimit points of Γv is P1pKvq ´ P1pKq.

Let us denote by ΓvzXv “ pΓvzXvq \ Ev Freudenthal’s compactification of ΓvzXv by itsfinite set of ends Ev, see [Fre]. This set of ends is indeed finite, in bijection with ΓvzP1pKqby the map which associates to z P ΓvzP1pKq the end towards which the cuspidal ray hzconverges. See for instance [Ser3] for a geometric interpretation of Ev in terms of the curveC.

Let Iv be the set of classes of fractional ideals of Rv. The map which associates to anelement rx : ys P P1pKq the class of the fractional ideal xRv ` yRv generated by x, y inducesa bijection from the set of cusps ΓzP1pKq to Iv.

The volume6 of the modular graph of groups can be computed using Equation (14.3) andExercice 2 b) in [Ser3, II.2.3]:

VolpPGL2pRvqzzXvq “ pq ´ 1qVolpGL2pRvqzzXvq “ 2 ζKp´1q . (15.7)

If K “ FqpY q is the rational function field over Fq and if we consider the valuation atinfinity v “ v8 of K, then the Nagao lattice7 Γv “ PGL2pFqrY sq acts transitively on P1pKq.Its quotient graph of groups ΓvzzXv is the following modular ray (with associated edge-indexedgraph)

Γ´1 Γ0 Γ1 Γ2

Γ10 Γ0 Γ1 Γ2

q ` 1 q 1 q 1 1 q 1q

where Γ´1 “ PGL2pFqq, Γ10 “ Γ0 X Γ´1 and, for every n P N,

Γn “

"„

a b0 d

P PGL2pFqrY sq : a, d P Fˆq , b P FqrY s,deg b ď n` 1

*

.

Note that even though PGL2pKvq has inversions on Xv, its subgroup Γv “ PGL2pRvq actswithout inversion on Xv (see for instance [Ser3, II.1.3]). In particular, the quotient graphΓvzXv is then well defined.

5See for instance [BasL] for a profusion of geometrically infinite lattices in simplicial trees.6See Section 2.7 for a definition.7This lattice was studied by Nagao in [Nag], see also [Moz, BasL]. It is called the modular group in [Wei2].

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15.3 Computations of measures for Bruhat-Tits trees

In this Section, we compute explicit expressions for the skinning measures of horoballs andgeodesic lines, and for the Bowen-Margulis measures, when considering lattices of Bruhat-Titstrees. See [PaP16a, Section 7] and [PaP16b, Section 4] for analogous computations in the realand complex hyperbolic spaces respectively, and [BrP1] for related computations in the treecase.

Let pKv, vq be as in the beginning of Section 15.1. Let Γ be a lattice of the Bruhat-Titstree Xv of pPGL2,Kvq. Since Xv is regular of degree qv ` 1, the critical exponent of Γ is

δΓ “ ln qv . (15.8)

We normalise the Patterson density pµxqxPV Xv of Γ as follows. Let H8 be the horoball inXv centred at 8 whose associated horosphere passes through ˚v. Let t ÞÑ xt be the geodesicray in Xv such that x0 “ ˚v and which converges to 8.

8

˚vBH8

Kv “ B8Xv ´ t8uπvOv

H8

xt

Ov

Hamenstädt’s measure8 associated with H8

µH8“ lim

tÑ`8eδΓtµxt “ lim

tÑ`8qvtµxt

is a Radon measure on B8Xv ´ t8u “ Kv, invariant under all isometries of Xv preservingH8, since Γ is a lattice. Hence it is invariant under the translations by the elements of Kv.By the uniqueness property of Haar measures, µH8

is a constant multiple of the chosen Haarmeasure9 of Kv, and we normalise the Patterson density pµxqxPV Xv so that

µH8“ HaarKv . (15.9)

We summarise the various measure computations in the following result.

Proposition 15.2. Let Γ be a lattice of the Bruhat-Tits tree Xv of pPGL2,Kvq, with Pattersondensity normalised as above.

8See Equation (7.5).9Recall that we normalise the Haar measure of pKv,`q such that HaarKv pOνq “ 1.

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(1) The outer/inner skinning measures of the singleton t˚vu are given by

d rσ˘t˚vu

pρq “ dµ˚vpρ˘q “ maxt1, |ρ˘|vu´2 dHaarKvpρ˘q

on the set of ρ P B1˘t˚vu such that ρ˘ ‰ 8.

(2) The total mass of the Patterson density is

µx “qv ` 1

qv

for all x P V Xv.

(3) The skinning measure of the horoball H8 is the projection of the Haar measure of Kv:For all ρ P B1

˘H8, we have

drσ˘H8pρq “ dµH8

pρ˘q “ dHaarKvpρ˘q .

(4) If 8 is a bounded parabolic fixed point of Γ, with Γ8 its stabiliser in Γ, if D “

pγH8qγPΓΓ8, we have

σ˘D “ HaarKvpΓ8zKvq “ VolpΓ8zzBH8q .

(5) Let L be a geodesic line in Xv with endpoints L˘ P Kv “ B8Xv ´ t8u. Then on the setof ρ P B1

`L such that ρ` P Kv “ B8Xv ´t8u and ρ` ‰ L˘, the outer skinning measureof L is

d rσ`L pρq “|L` ´ L´|v

|ρ` ´ L´|v |ρ` ´ L`|vdHaarKvpρ`q .

(6) Let L be a geodesic line in Xv, let ΓL be the stabiliser in Γ of L, and assume that ΓLzLhas finite length. Then with D “ pγLqγPΓΓL, we have

σ˘D “qv ´ 1

qvVolpΓLzzLq .

Proof. (1) For every ξ P Kv, by the description of the geodesic lines in the Bruhat-Tits treeXv starting from 8 given in Section 15.1, we have ξ P Ov if and only if PH8

pξq “ ˚v.10

8

˚v

Kv “ B8Xv ´ t8u

H8

ξ

pH8pξq

BH8

0

´vpξq

10Recall that PH8 : BvXv ´ t8u Ñ BH8 is the closest point map to the horoball H8, see Section 2.5.

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For every ξ P Kv ´ Ov, by Equations (15.2) and (2.8), we have

|ξ|v “ dH8p0, ξqln qv “ q

12dp˚v , PH8 pξqq

v . (15.10)

On the set of geodesic rays ρ P B1˘t˚vu such that ρ˘ ‰ 8, by Equation (7.2), by the last

claim of Proposition 7.2, by Equation (15.8),11 since PH8pρ˘q belongs to the geodesic ray

r˚v, ρ˘r (even when ρ˘ P Ov), and by Equation (15.9), we have

drσ˘t˚vu

pρq “ dµ˚vpρ˘q “ eCρ˘ pPH8 pρ˘q,˚vq dµH8pρ˘q

“ eδΓ βρ˘ pPH8 pρ˘q,˚vq dµH8pρ˘q

“ q´dpPH8 pρ˘q,˚vqv dHaarKvpρ˘q .

Therefore, if ρ P B1˘t˚vu is such that

ρ˘ P Ov “ tξ P Kv : |ξ|v ď 1u “ tξ P Kv : PH8pξq “ ˚vu ,

then d rσ˘t˚vu

pρq “ dHaarKvpρ˘q . If ρ˘ P Kv ´ Ov, Equation (15.10) gives the claim.

(2) This Assertion follows from Assertion (1) by a geometric series argument, but we give adirect proof.

As Γ is a lattice, the family pµxqxPV Xv is actually equivariant under AutpXvq,12 which actstransitively on the vertices of Xv, and the stabiliser in AutpXvq of the standard base point ˚vacts transitively on the edges starting from ˚v.

Since Xv is pqv ` 1q-regular, since the set of points at infinity of the geodesic rays startingfrom ˚v, whose initial edge has endpoint 0 P lkp˚vq “ P1pkvq, is equal to πvOv, since allgeodesic lines from 8 P B8Xv to points of πvOv Ă B8Xv pass through ˚v, and by thenormalisation of the Patterson density and of the Haar measure, we have

µ˚v “ pqv ` 1q µ˚vpπvOvq “ pqv ` 1q µH8pπvOvq “ pqv ` 1qHaarKvpπvOvq

“qv ` 1

qvHaarKvpOvq “

qv ` 1

qv.

(3) This follows from Equation (7.4), and from the normalisation µH8“ HaarKv of the

Patterson density.

(4) This follows from Assertion (3) and from Equation (8.10) (where the normalisation of thePatterson density was different).

11Since the potential is zero, the Gibbs cocycle is the critical exponent times the Busemann cocycle.12See Proposition 4.14 (2).

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ρp`8q

˚v

ρp0q

B8X

L`L´

(5) This follows from Lemma 8.6 applied with H “ H8, from Equations (15.9), (15.2) and(15.8).

(6) This follows from Assertion (2) and Equation (8.11) (where the normalisation of thePatterson density was different), since Xv is pqv ` 1q-regular:

σ˘D “ µ˚vqv ´ 1

qv ` 1VolpΓLzzLq “

qv ´ 1

qvVolpΓLzzLq . l

We now turn to measure computations for arithmetic lattices Γ in Xv in the function fieldcase. We still assume that the Patterson density of Γ is normalised so that µH8

pOvq “ 1, andwe denote by mBM the Bowen-Margulis measure of Γ associated with this choice of Pattersondensity.

Proposition 15.3. Let K be a function field over Fq and let v be a valuation of K. Let Γbe a finite index subgroup of Γv “ PGL2pRvq, with Patterson density normalised such thatµH8

“ HaarKv .

(1) We have

mBM “pqv ` 1q rΓv : Γs

qvVolpΓvzzXvq “

2 pqv ` 1q ζKp´1q rΓv : Γs

qv,

and if K “ FqpY q and v “ v8 is the valuation at infinity of C “ P1, then

mBM “2 rPGL2pFqrY sq : Γs

q pq ´ 1q2.

(2) Let Γ8 be the stabiliser in Γ of 8 P B8Xv, and let D “ pγH8qγPΓΓ8. We have

σ˘D “q g´1 rpΓvq8 : Γ8s

q ´ 1.

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Proof. (1) Recall that Γv “ PGL2pRvq acts without inversion on Xv. By Equation (8.3)(which uses a different normalisation of the Patterson density of Γ), and by Proposition 15.2(2), we have

mBM “ µ˚v2 qvqv ` 1

VolpΓzzXvq “pqv ` 1q rΓv : Γs

qvVolpΓvzzXvq .

The first claim of Assertion (1) hence follows from Equation (15.7).If K “ FqpY q and v “ v8, then the second claim of Assertion (1) follows either from the

first claim where the value of ζKp´1q is given by Equation (14.5), or from the fact that qv “ qand that the covolume VolpPGL2pFqrY sqzzXv8q of the Nagao lattice PGL2pFqrY sq is

VolpPGL2pFqrY sqzzXv8q “2

pq ´ 1qpq2 ´ 1q, (15.11)

as an easy geometric series computation shows using the description of the modular ray inSection 15.2 (see also [BasL, Sect. 10.2]).

(2) Let us prove that

HaarKvppΓvq8zKvq “q g´1

q ´ 1. (15.12)

The result then follows by Proposition 15.2 (4) since

σ˘D “ HaarKvpΓ8zKvq “ rpΓvq8 : Γ8s HaarKvppΓvq8zKvq .

The stabiliser of 8 “ r1 : 0s in Γv acts on Kv exactly by the set of transformationsz ÞÑ az` b with a P pRvqˆ and b P Rv. Since pRvqˆ “ pFqqˆ (see Equation (14.3)) acts freelyby left translations on pKv ´RvqRv, and by Lemma 14.4, we have

HaarKvppΓvq8zKvq “1

q ´ 1HaarKvpKvRvq “

q g´1

q ´ 1.

This proves Equation (15.12). l

15.4 Exponential decay of correlation and error terms for arith-metic quotients of Bruhat-Tits trees

As in the beginning of Section 15.1, let Kv be a non-Archimedean local field, with valuationv, valuation ring Ov, choice of uniformiser πv, and residual field kv of order qv. Let Γ be alattice of the Bruhat-Tits tree Xv of pPGL2,Kvq. In this Section, we discuss the error termsestimates that we will use in Part III.

In part in order to simplify the references, we start by summarizing in the next statementthe only results from the geometric Part II of this book, on geometric equidistribution andcounting problems, that we will use in this algebraic Part III. We state it with the normali-sation which will be useful there (see Section 15.3).

Theorem 15.4. Let Γ be a lattice of Xv whose length spectrum LΓ is equal to 2Z. Assumethat the Patterson density of Γ is normalised so that µx “ qv`1

qvfor every x P V Xv. Let D˘

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be nonempty proper simplicial subtrees of Xv with stabilisers ΓD˘ in Γ, such that the familiesD˘ “ pγD˘qγPΓΓD˘

are locally finite in Xv. If the measure σ´D` is nonzero and finite, then

limnÑ`8

pqv2 ´ 1qpqv ` 1q

2 qv3

VolpΓzzXvqσ´D`

q´nv

ÿ

γPΓΓD`0ădpD´, γD`qďn

∆α´e, γ“ rσ`D´ ,

for the weak-star convergence of measures on the locally compact spacep

GXv.Furthermore, if Γ is geometrically finite, then for every β P s0, 1s, there exists an error

term for this equidistribution claim when evaluated on rψ P C βc p

p

GXq of the form Op rψ β e´κnq

for some κ ą 0.

As recalled at the end of Section 2.7, arithmetic lattices in PGL2pKvq are geometricallyfinite, see [Lub1]. We will hence be able to use the error term in Theorem 15.4 in particularwhen‚ Kv is the completion of a function fieldK over Fq with respect to a (normalised discrete)

valuation v of K and Γ is a finite index subgroup of PGL2pRvq with Rv the affine functionring associated with v,13 as in Chapters 16 and 19, and in Sections 17.2 and 18.2;‚ when Kv “ Qp and Γ is an arithmetic lattice in PGL2pKvq derived from a quaternion

algebra, see Sections 17.3 and 18.2.

Proof. In order to prove the first claim, we apply Corollary 11.11 with X “ Xv and p “ q “ qv.Since LΓ “ 2Z, the lattice Γ leaves invariant the partition of V Xv into vertices at even distancefrom a base point x0 and vertices at odd distance from x0. Since the Patterson density is nownormalised so that µx0 “

qv`1qv

(instead of µx0 “qv`1?qv

in Corollary 11.11), the skinningmeasures σ¯D˘ are now 1?

qvtimes the ones in the statement of Corollary 11.11. Hence the

second assertion of Corollary 11.11 gives

limnÑ`8

qv2 ´ 1

2 qv2

TVolpΓzzXvq?qv σ

´

D`qv´n

ÿ

γPΓΓD`0ădpD´, γD`qďn

∆α´e, γ“?qv rσ`D´ .

By Equation (2.16), we have

TVolpΓzzXvq “ pqv ` 1q VolpΓzzXvq .

The first claim follows.

The last claim concerning error terms follows from Remark (ii) following the statement ofTheorem 12.17. l

In the last four Chapters 16, 17, 18 and 19 of this book, we will need to push to infinitythe measures appearing in the statement of Theorem 15.4. We regroup in the following twolemmas the necessary control tools for such a pushing.

The first one is a metric estimate on the extension of geodesic segments to geodesic rays.

Lemma 15.5. Let X be a geodesically complete proper CATp´1q space, let T ě 1, and let α Pp

GX be a generalised geodesic line which is isometric exactly on r0, T s. For every generalised13See Section 14.2 for definitions

237 19/12/2016

geodesic line ρ Pp

GX which is isometric exactly on r0,`8r , such that ρ|r0,T s “ α|r0,T s, wehave

dpα, ρq “e´2T

4ă 1 ,

and hence for all β P s0, 1s and rψ P C βb p

p

GXq,

| rψpαq ´ rψpρq| ďe´2β T

4β rψ β .

With the notation of Theorem 15.4, we will use this result whenX “ |Xv|1 is the geometricrealisation of the simplicial tree Xv, α “ α´e, γ is14 the common perpendicular between D´and γD` for γ P Γ (when it exists), and ρ “ ργ is any extension of α to a geodesic ray, orrather to a generalised geodesic line isometric exactly on r0,`8r . Under the assumptions ofTheorem 15.4, we have

pqv2 ´ 1qpqv ` 1q

2 qv3

VolpΓzzXvqσ´D`

q´nv

ÿ

γPΓΓD`0ădpD´, γD`qďn

∆ργ˚á rσ`D´ , (15.13)

with, if Γ is geometrically finite, an error term when evaluated on rψ P C βc p

p

GXq of the formOp rψ β e

´κnq for some κ ą 0 small enough (depending in particular on β P s0, 1s).

Proof. By Equation (2.4) defining the distance onp

GX, we have, since dpαptq, ρptqq “ 0 forall t P s ´8, T s and dpαptq, ρptqq “ t´ T otherwise,

dpα, ρq “

ż `8

Tpt´ T q e´2t dt “ e´2T

ż `8

0u e´2u du “

e´2T

4.

The result follows. l

The second lemma is a metric estimate on the map which associates to a geodesic ray itspoint at infinity. We start by giving some definitions.

Let X be a geodesically complete proper CATp´1q space, and let D be a nonempty properclosed convex subset of X. The distance-like map dD :

`

B8X ´ B8D˘2Ñ r0,`8r associated

with D is defined in [HeP4, §2.2] as follows: For ξ, ξ1 P B8X ´ B8D, let ξt, ξ1t : r0,`8r Ñ Xbe the geodesic rays starting at the closest points PDpξq, PDpξ1q to ξ, ξ1 on D and convergingto ξ, ξ1 as tÑ8. Let

dDpξ, ξ1q “ lim

tÑ`8e

12dpξt, ξ1tq´t . (15.14)

The distance-like map dD is invariant by the diagonal action of the isometries of X preservingD. If D consists of a single point x, then dD is the visual distance15 dx on B8X based at x. IfD is a horoball with point at infinity ξ0, then dD is Hamenstädt’s distance16 on B8X ´ tξ0u.As seen in [HeP4, §2.2, Ex. (4)], if X is a metric tree, then

dDpξ, ξ1q “

$

’

&

’

%

e12dpPDpξq, PDpξ

1qq ą 1 if PDpξq ‰ PDpξ1q

dxpξ, ξ1q “ e´dpx, yq ď 1 if PDpξq “ PDpξ

1q “ x

and rx, ξr X rx, ξ1r “ rx, ys .

14the generalised geodesic line isometric exactly on r0, dpD´, γD`qs parametrising15See Equation (2.1).16See Equation (2.8).

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In particular, although it is not an actual distance on its whole domain B8X´B8D, the mapdD is locally a distance, and we can define with the standard formula the β-Hölder-continuityof maps with values in pB8X ´ B8D, dDq and the β-Hölder-norm of a function defined onpB8X ´ B8D, dDq. From now on, we endow B8X ´ B8D with the distance-like map dD.

Proposition 15.6. Let X be a locally finite simplicial tree without terminal vertices, and let Dbe a proper nonempty simplicial subtree of X. The homeomorphism B` : B1

`DÑ pB8X´B8Dqdefined by ρ ÞÑ ρ` is 1

2 -Hölder-continuous, and for all β P s0, 1s and ψ P C βb pB8X ´ B8Dq,

the map ψ ˝ B` : B1`DÑ R is bounded and β

2 -Hölder-continuous, with

ψ ˝ B`β2ď p1` 2

β2`1q ψβ .

With the notation of Theorem 15.4, using the claim following the statement of Lemma15.5, we will use this result when X “ Xv and D “ D´. Under the assumptions of Theorem15.4, with ργ any extension to a geodesic ray of the common perpendicular α´e, γ between D´and γD` for γ P Γ, since pushing forward measures on B1

`D´ by the homeomorphism B` iscontinuous, we have by Equation (15.13)

pqv2 ´ 1qpqv ` 1q

2 qv3

VolpΓzzXvqσ´D`

q´nv

ÿ

γPΓΓD`0ădpD´, γD`qďn

∆pργq`˚á pB`q˚rσ

`

D´ . (15.15)

If Γ is geometrically finite, for all β P s0, 1s and ψ P C βc pB8Xv ´ B8D´q, using the error term

in Equation (15.13) with regularity β2 when evaluated on rψ “ ψ ˝ B` P C

β2c p

p

GXvq, we haveby the last claim of Proposition 15.6 an error term in Equation (15.15) evaluated on ψ of theform Op ψ β e

´κnq for some κ ą 0 small enough.

Proof. Let us prove that for every ρ, ρ1 P B1`D, if dpρ, ρ1q ď 1, then ρp0q “ ρ1p0q, and

dDpρ`, ρ1`q “

?2 dpρ, ρ1q

12 . (15.16)

This proves that the map B` is 12 -Hölder-continuous. We may assume that ρ ‰ ρ1.

Let ρ, ρ1 P B1`D. If ρp0q ‰ ρ1p0q, then the images of ρ and ρ1 are disjoint and their

connecting segment in the tree X joins ρp0q and ρ1p0q; hence for every t P r0,`8r , we have

dpρptq, ρ1ptqq “ dpρp0q, ρ1p0qq ` dpρptq, ρp0qq ` dpρ1ptq, ρ1p0qq ě 1` 2 t .

Thus

dpρ, ρ1q “

ż 0

´8

dpρp0q, ρ1p0qq e2 t dt`

ż `8

0dpρptq, ρ1ptqq e´2 t dt

ě

ż 0

´8

e2 t dt`

ż `8

0p1` 2tq e´2 t dt ą 2

ż `8

0e´2 t dt “ 1 .

Assume that x “ ρp0q “ ρ1p0q and let n be the length of the intersection of ρ and ρ1. Then

dDpρ`, ρ1`q “ dxpρ`, ρ

1`q “ lim

tÑ`8e

12dpρptq, ρ1ptqq´t “ e´n .

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Furthermore, since ρptq “ ρ1ptq for t ď n and dpρptq, ρ1ptqq “ 2pt ´ nq otherwise, we have,using the change of variables u “ 2pt´ nq,

dpρ, ρ1q “

ż `8

n2 pt´ nq e´2 t dt “ e´2n

ż `8

0u e´u

du

2“e´2n

2.

This proves Equation (15.16).Let β P s0, 1s and ψ P C β

b pB8X ´ B8Dq. We have ψ ˝ B`8 “ ψ8 since B` is ahomeomorphism, and, by Equation (15.16),

ψ ˝ B`1β2

“ supρ, ρ1PB1

`D, 0ădpρ, ρ1qď1

|ψ ˝ B`pρq ´ ψ ˝ B`pρ1q|

dpρ, ρ1qβ2

ď supρ, ρ1PB1

`D, 0ădpρ, ρ1qď 12

|ψ ˝ B`pρq ´ ψ ˝ B`pρ1q|

dpρ, ρ1qβ2

`2 ψ ˝ B`8

2´β2

ď supξ, ξ1PB8X´B8D0ădDpξ, ξ

1qď1

|ψpξq ´ ψpξ1q|

2´β2 dDpξ, ξ1qβ

` 2β2`1 ψ8

ď 2β2`1 ψβ .

Since ψ ˝ B`β2“ ψ ˝ B`8 ` ψ ˝ B

`1β2

, this proves the last claim of Proposition 15.6. l

We conclude Section 15.4 by giving a purely algebraic control of error terms, under thestronger regularity requirement on functions of being locally constant. We assume until theend of this Section that the lattice Γ is contained in Gv “ PGL2pKvq.

The group Gv acts (on the left) on the complex vector space of maps ψ from ΓzGv to R,by right translation on the source: For every g P Gv, we have gψ : x ÞÑ ψpxgq. A functionψ : ΓzGv Ñ R is algebraically locally constant if there exists a compact-open subgroup U ofPGL2pOvq which leaves ψ invariant:

@ g P U, gψ “ ψ ,

or equivalently, if ψ is constant on each orbit of U under the right action of Gv on ΓzGv. Notethat ψ is then continuous, since the orbits of U are compact-open subsets. We define

dψ “ dim`

VectRpPGL2pOvqψq˘

as the dimension of the complex vector space generated by the images of ψ under the elementsof PGL2pOvq, which is finite, and even satisfies

dψ ď rPGL2pOvq : U s .

We define the alc-norm of every bounded algebraically locally constant map ψ : ΓzGv Ñ Rby

ψalc “a

dψ ψ8 .

Though the alc-norm does not satisfy the triangle inequality, we have λψalc “ |λ| ψalc

for every λ P R. We denote by alcpΓzGvq the vector space of bounded algebraically locallyconstant maps ψ from ΓzGv to R.

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For every n P N, let Un be the compact-open subgroup of PGL2pOvq which is the kernelof the morphism PGL2pOvq Ñ PGL2pOvπv

nOvq of reduction modulo πvn. Note that anycompact-open subgroup U of PGL2pOvq contains Un for some n P N. Hence ψ : ΓzGv Ñ R isalgebraically locally constant if and only if there exists n P N such that ψ is constant on eachright orbit of Un. For every n P N, since the order of PGL2pOvπv

nOvq is at most the orderof pOvπv

nOvq4, which is qv4n, if ψ : ΓzGv Ñ R is constant on each right orbit of Un, then

ψalc ď qv2n ψ8 . (15.17)

The next result is an algebraic version of the error term statement in Theorem 15.4(assuming for simplicity that LΓ “ Z), which uses17 a stronger assumption on Γ, and obtains aweaker regularity (locally constant instead of Hölder-continuous, see Section 3.1 for definitionsand notation). We will not use it in this book, but its version with LΓ “ 2Z is used in theannouncement [BrPP] which only considers the locally constant regularity.

Theorem 15.7. Let Kv be the completion of a function field K over Fq with respect to avaluation v of K and let Γ be a nonuniform lattice of Gv with LΓ “ Z. Then there existsκ ą 0 such that for every ε P s0, 1s and every ε-locally constant map rψ :

p

GXv Ñ R, we have,as nÑ `8,

pqv ´ 1q

pqv ` 1q qv

VolpΓzzXvqσ´D`

q´nv

ÿ

γPΓΓD`0ădpD´, γD`qďn

rψpα´e, γq

“

ż

p

GXv

rψ drσ`D´ `Op rψ ε lc, ln qv e´κnq .

Proof. This result follows by replacing in the proof of Theorem 12.17 (or rather Remark (ii)following its statement) the use of the exponential decay of β-Hölder correlations given byCorollary 9.6 by the following result of decay of correlations under locally constant regularity(which does follow from Corollary 9.6 by Remark 3.2).

Proposition 15.8. Let Kv be the completion of a function field K over Fq with respect toa valuation v of K and let Γ be a nonuniform lattice of Gv with LΓ “ Z. Then there existC, κ ą 0 such that for every ε P s0, 1s, for all ε-locally constant maps φ, ψ : ΓzGXv Ñ R andn P Z, we have

ˇ

ˇ

ˇ

ż

ΓzGXvφ ˝ g´n ψ dmBM ´

1

mBM

ż

ΓzGXvφ dmBM

ż

ΓzGXvψ dmBM

ˇ

ˇ

ˇ

ď C e´κ|n| φε lc, ln qv ψε lc, ln qv .

Proof. Recall18 that we have a natural homeomorphism Ξ : ΓgApOvq ÞÑ Γg `˚ betweenΓzGvApOvq and ΓzGXv. We denote by pG : ΓzGv Ñ ΓzGXv the composition map of thecanonical projection

`

ΓzGv˘

Ñ`

ΓzGvApOvq˘

and of Ξ. By Equation (15.4), for everyx P ΓzGv, we have

pG px avnq “ gn pG pxq . (15.18)

17Note that the existence of a nonuniform lattice in Gv “ PGL2pKvq forces the characteristic to be positive,see for instance [Lub1].

18See Section 15.1.

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Lemma 15.9. For every ε P s0, 1s, for every ε-locally constant function ψ : ΓzGXv Ñ R, ifn “ r´1

2 ln εs, then the map ψ ˝ pG : ΓzGv Ñ R is Un-invariant and

ψ ˝ pG alc ď qv2 ψε lc, ln qv . (15.19)

Proof. Let ε, ψ, n be as in the statement. Let us first prove that if `, `1 P GXv satisfy`r´n,`ns “ `1

r´n,`ns, then dp`, `1q ď ε.

If `r´n,`ns “ `1r´n,`ns, then dp`ptq, `

1ptqq “ 0 for t P r´n, ns and by the triangle inequalitydp`ptq, `1ptqq ď 2p|t| ´ nq if |t| ě n, hence

dp`, `1q ď 2

ż `8

n2 pt´ nq e´2 t dt “ 2 e´2n

ż `8

0u e´u

du

2“ e´2n

ď e´2p´ 12

ln εq “ ε ,

as wanted.

In order to prove that ψ ˝ pG : ΓzGv Ñ R is Un-invariant, let x, x1 P ΓzGv be such thatx1 P xUn. Since Un acts by the identity map on the ball of radius n in the Bruhat-Tits tree Xv,the geodesic lines pG pxq and pG px

1q in ΓzGXv coincide (at least) on r´n, ns. Hence, as we sawin the beginning of the proof, we have dppG pxq, pG px

1qq ď ε. Therefore ψppG pxqq “ ψppG px1qq

since ψ is ε-locally constant.Now, using Equation (15.17), we have

ψ ˝ pG alc ď qv2n ψ ˝ pG 8

ď qv2p1´ 1

2ln εq ψ8 “ qv

2 ε´ ln qv ψ8 “ qv2 ψε lc, ln qv . l

Now, in order to prove Proposition 15.8, we will use an algebraic result of exponential decayof correlations, Theorem 15.10 (see for instance [AtGP]). We first recall some definitions andnotation, useful for its statement.

Recall that the left action of the locally compact unimodular group Gv on the locallycompact space GXv is continuous and transitive, and that its stabilisers are compact henceunimodular. Since Γ is a lattice, the (Borel, positive, regular) Bowen-Margulis measurermBM on GXv is Gv-invariant (see Proposition 4.14 (2)). Hence by [Wei1] (see also [GoP,Lem. 5]), there exists a unique Haar measure on Gv, which disintegrates by the evaluation mapĂpG : Gv Ñ GXv defined by g ÞÑ g`˚, with conditional measure on the fiber over ` “ g`˚ P GXvthe probability Haar measure on the stabiliser gApOvqg

´1 of ` under Gv. Hence, taking thequotient under Γ and normalising in order to have probability measures, if µv is the rightGv-invariant probability measure on ΓzGv, we have

ppG q˚µv “mBM

mBM. (15.20)

For every g P Gv, we denote by Rg : ΓzGv Ñ ΓzGv the right action of g, and for allbounded continuous functions rψ, rψ1 on ΓzGv, we define

covµv , gprψ, rψ1q “

ż

ΓzGv

rψ ˝Rg rψ1 dµv ´

ż

ΓzGv

rψ dµv

ż

ΓzGv

rψ1 dµv .

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Note that by Equations (15.20) and (15.18), for all bounded continuous functions ψ,ψ1 :ΓzGXv Ñ R and n P Z, we have19

cov mBMmBM

, npψ,ψ1q “ covµv , avnpψ ˝ pG , ψ

1 ˝ pG q . (15.21)

Recall that the adjoint representation of Gv “ PGL2pKvq is the continuous morphismGv Ñ GLpM2pKvqq defined by rhs ÞÑ tx ÞÑ hxh´1u, which is independent of the choice ofthe representative h P GL2pKvq of rhs P PGL2pKvq. For every g P Gv, we denote by |g|v the

operator norm of the adjoint representation of g. For instance, recalling that av “„

1 00 π´1

v

,

we have, for all n P Z,|av

n|v “ qv|n| . (15.22)

We refer for instance to [AtGP] for the following result of exponential decay of correlations.

Theorem 15.10. Let Γ be a nonuniform lattice of Gv. There exist C 1, κ1 ą 0 such that, forall bounded locally constant functions rψ, rψ1 : ΓzGv Ñ R and g P Gv,

ˇ

ˇ covµv , gprψ, rψ1q

ˇ

ˇ ď C 1 rψalc rψ1alc |g|v

´κ1 . l (15.23)

Proposition 15.8 follows from this result applied to rψ “ ψ ˝ pG , rψ1 “ ψ1 ˝ pG and g “ avn

by using Equations (15.21), (15.19) and (15.22) and by taking C “ C 1q4v and κ “ κ1 ln qv. l

This concludes the proof of Theorem 15.7. l

Remark. There is a similar relationship between locally constant functions on Kv in analgebraic sense and the ones in the metric sense.

The additive group pKv,`q acts on the complex vector space of functions from Kv to R,by translations on the source: for all y P Kv and ψ : Kv Ñ R, the function y ¨ ψ is equal tox ÞÑ ψpx ` yq. A map ψ : Kv Ñ R is algebraically locally constant if there exists k P N suchthat ψ is invariant under the action of the compact-open subgroup pπvqkOv of Kv, that is, iffor all x P Kv and y P pπvqkOv, we have ψpx` yq “ ψpxq. Note that a locally constant mapfrom Kv to R is continuous.

For any locally constant function ψ : Kv Ñ R, the complex vector space VectRpOv ¨ ψqgenerated by the images of ψ under the elements of Ov is finite dimensional. Its dimensiondψ satisfies, with k as above,

dψ ď rOv : pπvqkOvs “ qv

k .

We define the alc-norm of every bounded algebraically locally constant map ψ : Kv Ñ R by

ψalc “a

dψ ψ8 .

Though the alc-norm does not satisfy the triangle inequality, we have λψalc “ |λ| ψalc forevery λ P R, and the set of bounded algebraically locally constant maps from Kv to R is areal vector space.

Actually, a function ψ : Kv Ñ R is algebraically locally constant if and only if it is locallyconstant. More precisely, for every ε P s0, 1s, since the closed balls of radius q´kv in Kv arethe orbits by translations under pπvqkOv, every ε-locally constant function ψ : Kv Ñ R isconstant under the additive action of pπvqkOv for k “ r´ ln ε

ln qvs, hence 20

ψalc ď ψε lc, 12.

19See Section 9.2 for a definition of covµ, n.20See Section 3.1 for the notation.

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15.5 Geometrically finite lattices with infinite Bowen-Margulismeasure

This Section is a digression from the theme of arithmetic applications, in which we use theNagao lattice defined in Section 15.2 in order to construct a geometrically finite discretegroup of automorphisms of a simplicial tree which has infinite Bowen-Margulis measure. Thisexample was promised towards the end of Section 4.4.

We will equivariantly change the lengths of the edges of a simplicial tree X endowed witha geometrically finite (nonuniform) lattice Γ in order to turn it into a metric tree in which thegroup Γ remains a geometrically finite lattice, but now has a geometrically finite subgroupwith infinite Bowen-Margulis measure. This example is an adaptation of the negatively curvedmanifold example of [DaOP, §4]. The simplicial example is obtained as a modification of themetric tree example.

Theorem 15.11. There exists a geometrically finite discrete group of automorphisms of aregular metric tree with infinite Bowen-Margulis measure.

There exists a geometrically finite discrete group of automorphisms of a simplicial treewith uniformly bounded degrees whose Bowen-Margulis measure is infinite.

Proof. Let pK “ FqppY ´1qq, O “ FqrrY ´1ss and R “ FqrY s. Let X be the Bruhat-Titstree of pPGL2, pKq with base point ˚ “ rO ˆ Os. Let Γ “ PGL2pRq, which is a lattice ofX, with quotient the modular ray ΓzzX described in Section 15.2. We denote by pyiqi“´1,0,...

the ordered vertices along ΓzzX with vertex stabilisers pΓiqi“´1,0,..., and by peiqiPN the orderededges along ΓzzX (pointing away from the origin of the modular ray).

The subgroup

P “ď

iě0

Γi “

"„

a Q0 d

: Q P FqrY s, a, d P Fˆq*

is the stabiliser in Γ of 8 P B8X. Let P0 be the finite index subgroup of P consisting of the

elements„

1 Q0 1

with Qp0q “ 0. Observing that dpγ˚, ˚q “ 2pi` 1q for any γ P Γi´Γi´1 and

that the cardinality of pΓi´Γi´1qXP0 is pq´ 1qqi`1, it is easy to see that the Poincaré series

QP, 0,˚,˚psq “ÿ

γPP

e´s dp˚,γ˚q

of the discrete (though elementary) subgroup P of IsompXq is (up to a multiplicative constant)equal to

ř8i“0 q

ie´2si, which gives δP “ ln q2 for the critical exponent of P on X.

Let h be a loxodromic element of Γ whose fixed points belong to the open subset Y ´1Oof pK “ B8X´ t8u. Hence the horoball H8 centred at 8 P B8X, whose horosphere contains˚, is disjoint from the translation axis Axh of h. Note that the stabiliser of H8 in Γ is Pand that P0 acts freely on the edges exiting H8. Let x0 P V X be the closest point on Axhto H8, let e˚ be the edge with origin ˚ pointing towards x0, and let e´, e` be the two edgeswith origin x0 on Axh.

244 19/12/2016

Uh

Axh

˚

Uh

e`e´

e˚

UP0

8

H8

x0

Let Uh be the set of points x in V X ´ tx0u such that the geodesic segment from x0 to xstarts either by the edge e´ or by e`. Let UP0 be the set of points y in V X ´ ttpe˚qu suchthat the geodesic segment from tpe˚q to y starts by the edge e˚. We have

(1) Uh X UP0 “ H and x0 R Uh Y UP0 ,

(2) hkpV X´Uhq Ă Uh for every k P Z´t0u and wpV X´UP0q Ă UP0 for every w P P0´tidu,

(3) dpx, yq “ dpx, x0q ` dpx0, yq for all x P Uh and y P UP0 .

Let Γ1 be the subgroup of Γ generated by P0 and h. By a ping-pong argument, Γ1 is afree product of P0 and of the infinite cyclic group generated by h. Hence every element γin Γ1 ´ teu may be written uniquely w0h

n0w1hn1 . . . wkh

nk with k P N, wi P P0, ni P Z withwi ‰ e if i ‰ 0 and ni ‰ 0 if i ‰ k. Using the above properties, we have by induction

dpx0, w0hn0w1h

n1 . . . wkhnkx0q “

ÿ

0ďiďk

dpx0, hnix0q `

ÿ

0ďiďk

dpx0, wix0q . (15.24)

Let λ : EX Ñ R` be the Γ1-invariant length map on the set of edges of X such that forevery i P N, the length of e P EX is 1 if e is not contained in

Ť

γPΓ1 γH8, and otherwise, if emaps to ei or to ei under the canonical map X Ñ ΓzX, then λpeq “ 1 ` ln i`1

i if i ě 1 andλpeq “ 1 if i “ 0. Note that the distance in the metric graph |X|λ from ˚ to the vertex onthe geodesic ray from ˚ to 8 originally at distance i from ˚ is now i ` ln i. The distancesalong the translation axis of h have not changed. Equation (15.24) remains valid with thenew distance.

Let us now prove that the discrete subgroup Γ1 of automorphisms of the regular metrictree pX, λq satisfies the first claim of Theorem 15.11.

By Γ1-invariance of λ, the group Γ1 remains a subgroup of AutpX, λq. The elements ofΓ18, or equivalently, the points at infinity of the horoballs in the Γ1-equivariant family ofhoroballs pγH8qγPΓ1Γ18 with pairwise disjoint interiors in X, remain bounded parabolic fixedpoints of Γ1, and the other limit points remain conical limit points of Γ1. Hence Γ1 remains ageometrically finite discrete subgroup of AutpX, λq.

The Poincaré series of the action of P0 on pX, λq is (up to a multiplicative and additiveconstant)

ř8i“1 q

i e´2si i´2s, which has the same critical exponent δP “ ln q2 as previously, but

it is easy to see that the group P is now of convergence type if q ě 3.

245 19/12/2016

The computations of [DaOP, §4] now apply to our situation (with C “ 0 in their notation,and we sum over P0 instead of over the infinite cyclic group generated by the parabolicelement p in their notation). Their argument shows that Γ1 is of convergence type with criticalexponent δP , up to replacing h by a big enough power. By Corollary 4.6, the Bowen-Margulismeasure of Γ1 is infinite.

In order to prove the second claim of Theorem 15.11, we first define a new length mapλ : EX Ñ R` which coincides with the previous one on every edge e of X, unless e maps toei or to ei under the canonical map XÑ ΓzX, in which case we set

λpeq “ 1` tlnpi` 1qu´ tln iu

if i ě 1 and λpeq “ 1 if i “ 0 (where t¨u is the largest previous integer map). This map λnow has values in t1, 2u, and we subdivide each edge of length 2 into two edges of length 1.The tree Y thus obtained has uniformly bounded degrees (although it is no longer a uniformtree), and the group Γ1 defines a geometrically finite discrete subgroup of AutpYq with infiniteBowen-Margulis measure. l

246 19/12/2016

Chapter 16

Rational point equidistribution andcounting in completed function fields

Let K be a function field over Fq, let v be a (normalised discrete) valuation of K, and letRv be the affine function ring associated with v. In this Chapter, we prove analogues of theclassical results on the counting and equidistribution towards the Lebesgue measure on Rof the Farey fractions p

q with pp, qq P Z ˆ pZ ´ t0uq relatively prime (see for instance [Nev],as well as [PaP14b] for an approach using methods similar to the ones in this text). Inparticular, we prove various equidistribution results of locally finite families of elements of Ktowards the Haar measure on Kv, using the geometrical work on equidistribution of commonperpendiculars done in Section 11.4 and recalled in Section 15.4.

16.1 Equidistribution of non-Archimedian Farey fractions

The first result of this Section is an analog in function fields of the equidistribution of Fareyfractions to the Lebesgue measure in R, see the Introduction, and for example [PaP14b, p. 978]for the precise statement and a geometric proof. For every px0, y0q P Rv ˆRv ´ tp0, 0qu, let

mv, x0, y0 “ Cardta P pRvqˆ : D b P x0Rv X y0Rv, pa´ 1qx0y0 ´ bx0 P y

20Rvu .

For future use, note that by Equation (14.3)

mv, 1, 0 “ q ´ 1 . (16.1)

For every pa, bq P Rv ˆ Rv and every subgroup H of GL2pRvq, let Hpa,bq be the stabiliser ofpa, bq for the linear action of H on Rv ˆRv.

Theorem 16.1. Let G be a finite index subgroup of GL2pRvq, and let px0, y0q P Rv ˆ Rv ´tp0, 0qu. Then as sÑ `8, if

c “pqv

2 ´ 1q pqv ` 1q ζKp´1q mv, x0, y0 pNxx0, y0yq2 rGL2pRvq : Gs

pq ´ 1q q g´1 q3v rGL2pRvqpx0,y0q : Gpx0,y0qs

,

thenc s´2

ÿ

px, yqPGpx0, y0q, |y|vďs

∆xy

˚á HaarKv .

247 19/12/2016

For every β P s0, 1ln qv

s, there exists κ ą 0 such that for every β-Hölder-continuous functionψ : Kv Ñ R with compact support,1 as for instance if ψ : Kv Ñ R is locally constant withcompact support (see Remark 3.2), there is an error term in the equidistribution claim ofTheorem 16.1 evaluated on ψ, of the form Ops´κψβq.

It is remarkable that due to the general nature of our geometrical tools, we are able to workwith any finite index subgroup G of GL2pRvq, and not only with its congruence subgroups. Inthis generality, the usual techniques (for instance involving analysis of Eisenstein series) arenot likely to apply. Also note that the Hölder regularity for the error term is a much weakerassumption than the locally constant one that is usually obtained by analytic number theorymethods.

Theorem 1.13 in the Introduction follows from this result, by taking K “ FqpY q (so thatg “ 0), v “ v8 and px0, y0q “ p1, 0q, and by using Equations (14.5) and (16.1) in order tosimplify the constant.

Before proving Theorem 16.1, let us give a counting result which follows from this equidis-tribution result by considering the locally constant characteristic function of a closed andopen fundamental domain of Kv modulo the action by translations of a finite index additivesubgroup of Rv, and by using Lemma 14.4 with I “ Rv.

The additive group Rv acts on Rv ˆRv by the horizontal shears (transvections):

@ k P Rv, @ px, yq P Rv ˆRv, k ¨ px, yq “ px` ky, yq ,

and this action preserves the absolute value |y|v of the vertical coordinate y. We may thendefine a counting function ΨG, x0,y0 of elements in K in an orbit by homographies under afinite index subgroup G of GL2pRvq, as

ΨG, x0, y0psq “ Card Rv,G z

px, yq P Gpx0, y0q, |y|v ď su ,

where Rv,G is the finite index additive subgroup of Rv consisting of the elements x P Rv such

thatˆ

1 x0 1

˙

P G. Note that Rv,G “ Rv if G “ GL2pRvq.

Corollary 16.2. Let G be a finite index subgroup of GL2pRvq, and let px0, y0q P Rv ˆ Rv ´tp0, 0qu. Then there exists κ ą 0 such that, as sÑ `8,

ΨG, x0,y0psq

“pq ´ 1q q 2g´2 q3

v rGL2pRvqpx0,y0q : Gpx0,y0qs rRv : Rv,Gs

pqv2 ´ 1q pqv ` 1q ζKp´1q mv, x0, y0 pNxx0, y0yq2 rGL2pRvq : Gs

s2 `Ops2´κq . l

Let us fix some notation for this Section. For every subgroup H of GL2pRvq, we denoteby H its image in Γv “ PGL2pRvq. Let Xv be the Bruhat-Tits tree of pPGL2,Kvq, which isregular of degree qv ` 1. Let

r “x0

y0P K Y t8u .

If y0 “ 0, let gr “ id P GL2pKq, and if y0 ‰ 0, let

gr “

ˆ

r 11 0

˙

P GL2pKq .

1where Kv is endowed with the distance px, yq ÞÑ |x´ y|v

248 19/12/2016

In the proof of Theorem 16.1, we apply Theorem 15.4 with Γ :“ G, D´ :“ H8 and D` :“grH8. Recall that H8 is the horoball in Xv centred at 8 whose boundary contains ˚v (seeSection 15.3).

Proof of Theorem 16.1. Note that Γ has finite index in Γv and, in particular, it is alattice of Xv. By [Ser3, II.1.2, Coro.], for all x P V Xv and γ P GL2pRvq, the distance dpx, γxqis even since vpdet γq “ 0. Hence by the equivalence in Equation (4.13), the length spectrumLΓv of Γv is 2Z. The length spectrum of Γ is also 2Z, since it is contained in LΓv .

Note that D` is a horoball in Xv centred at r “ x0y0P B8Xv, by Lemma 15.1. The stabiliser

ΓD´ of D´ (respectively ΓD` of D`) coincides with the fixator Γ8 of 8 P B8Xv (respectivelythe fixator Γr of r) in Γ. Note that the families D˘ “ pγD˘qγPΓΓD˘

are locally finite, sinceΓv, and hence its finite index subgroup Γ, is geometrically finite,2 and since 8 and r P K arebounded parabolic limit points of Γv, hence of its finite index subgroup Γ.

For every γ P ΓΓr such that D´ and γD` are disjoint, let ργ be the geodesic ray startingfrom α´e, γp0q and ending at the point at infinity γ ¨ r of γD`. Note that ργ and α´e, γ coincideon r0, dpD´, γD`qs.

Since the Patterson densities of lattices of Xv have total mass qv`1qv

by Proposition 15.2(2), they are normalized as in Theorem 15.4. Then by Equation (15.15), we have

limnÑ`8

pqv2 ´ 1qpqv ` 1q

2 q3v

VolpΓzzXvqσ´D`

qv´n

ÿ

γPΓΓr0ădpD´, γD`qďn

∆pργq` “ pB`q˚rσ`

D´ . (16.2)

Furthermore, for every β P s0, 1ln qv

s, by the comment following Equation (15.15), we have anerror term of the form Opψβ ln qve

´κnq for some κ ą 0 in the above formula when evaluatedon ψ P C β ln qv

c pBXv ´ t8uq, where BXv ´ t8u is endowed with Hamenstädt’s distance dH8.

Hence we have an error term Opψβe´κnq for some κ ą 0 in the above formula when evaluated

on ψ P C βc pKvq, where Kv “ BXv ´ t8u is endowed with the distance px, yq ÞÑ |x ´ y|v, see

Equation (15.3).By Proposition 15.2 (3), we have

pB`q˚rσ`

D´ “ HaarKv .

Hence Equation (16.2) gives, with the appropriate error term,

limnÑ`8

pqv2 ´ 1qpqv ` 1q

2 q3v

VolpΓzzXvqσ´D`

qv´n

ÿ

γPΓΓr0ădpD´, γD`qďn

∆γ¨r “ HaarKv . (16.3)

Let g P GL2pKq be such that g8 ‰ 8. This condition is equivalent to asking that thep2, 1q-entry c of g is nonzero. By Lemma 15.1, the signed distance between the horospheresH8 and gH8 is

dpH8, gH8q “ ´2 vpcq “ 2ln |c|vln qv

. (16.4)

If y0 ‰ 0, then px, yq “ gpx0, y0q if and only if p xy0, yy0q “ ggrp1, 0q, and the p2, 1q-entry of ggr

is yy0. If y0 “ 0 (which implies that gr “ id and x0 ‰ 0), then px, yq “ gpx0, y0q if and only if

2See Section 15.2

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p xx0, yx0q “ gp1, 0q, and the p2, 1q-entry of g “ ggr is y

x0. Let

z0 “

"

y0 if y0 ‰ 0x0 otherwise.

By Equation (16.4), the signed distance between D´ “ H8 and gD` “ g grH8 is

dpD´, gD`q “2

ln qvlnˇ

ˇ

y

z0

ˇ

ˇ

v.

By discreteness, there are only finitely many double classes rgs P Gp1,0qzGGpx0,y0q suchthat D´ “ H8 and gD` “ g grH8 are not disjoint. Let ZpGq be the centre of G, whichis finite. Since ZpGq acts trivially on P1pKvq, the map GGpx0,y0q Ñ ΓΓr induced by thecanonical map GL2pRvq ÞÑ PGL2pRvq is |ZpGq|-to-1. Using the change of variable

s “ |z0|v qvn2 ,

so that qv´n “ |z0|v2 s´2, Equation (16.3) gives, with the appropriate error term,

limsÑ`8

pqv2 ´ 1q pqv ` 1q |z0|v

2

2 qv3 |ZpGq|

VolpΓzzXvqσ´D`

s´2ÿ

px, yqPGpx0, y0q, |y|vďs

∆xy“ HaarKv . (16.5)

The order of the centre ZpGL2pRvqq “ pRvqˆ id is q ´ 1 by Equation (14.3). The map

GL2pRvqGÑ ΓvΓ induced by the canonical map GL2pRvq Ñ PGL2pRvq is hence q´1|ZpGq| -to-1.

By Equation (15.7), we hence have

VolpΓzzXvq “ rΓv : ΓsVolpΓvzzXvq “ 2 ζKp´1q rΓv : Γs

“2

q ´ 1ζKp´1q |ZpGq| rGL2pRvq : Gs . (16.6)

Theorem 16.1 follows from Equations (16.5) and (16.6) and from Lemma 16.3 below. l

Lemma 16.3. We have

σ´D` “q g´1 |z0|v

2rGL2pRvqpx0,y0q : Gpx0,y0qs

mv, x0, y0 pNxx0, y0yq2

.

Proof. Let γr be the image of gr in PGL2pKq. Let us define Γ1 “ γr´1Γγr, which is a finite

index subgroup in Γ1v “ γr´1Γvγr and a lattice of Xv. Since γr maps 8 to r, the point 8 is a

bounded parabolic limit point of Γ1, and we have pΓ1q8 “ γr´1Γrγr. Since the canonical map

GL2pRvq Ñ PGL2pRvq is injective on the stabiliser GL2pRvqpx0,y0q, we have

rpΓ1vq8 : pΓ1q8s “ rpΓvqr : Γrs “ rGL2pRvqpx0,y0q : Gpx0,y0qs .

Since the Patterson density of a lattice does not depend on the lattice (see Proposition4.14 (1)), the skinning measures rσ˘H of a given horoball H do not depend on the lattice.Thus

γ˚ rσ˘H “ rσ˘γH

250 19/12/2016

for every γ P AutpXvq. Let D`1 “ pγ1H8qγ1PΓ1Γ18 , which is a locally finite Γ1-equivariant

family of horoballs. We hence have, using Proposition 15.2 (4) for the third equality,

σ´D` “ σ´

γrD`1

“ σ´D`1 “ HaarKvppΓ

1q8zKvq

“ rpΓ1vq8 : pΓ1q8s HaarKvppΓ1vq8zKvq

“ rGL2pRvqpx0,y0q : Gpx0,y0qs HaarKvppΓ1vq8zKvq . (16.7)

Every element in the stabiliser of 8 in PGL2pKvq can be uniquely written in the form

α “

„

a b0 1

with pa, bq P pKvqˆ ˆKv. Note that

ˆ

r 11 0

˙ˆ

a b0 1

˙ˆ

0 11 ´r

˙

“

ˆ

br ` 1 ar ´ br2 ´ rb a´ br

˙

.

When x0 “ 0 or y0 “ 0, we have α P Γ1v if and only if

b P Rv and a P pRvqˆ .

When x0, y0 ‰ 0, we have α P Γ1v if and only if γrαγr´1 P Γv, hence if and only if

b P Rv X1

rRv, a P pRvq

ˆ, ar ´ br2 ´ r P Rv .

Let U 18 be the kernel of the map from pΓ1vq8 to pKvqˆ sending

„

a b0 1

to a, and let mv

be its index in pΓ1vq8. If x0 “ 0 or y0 “ 0, then mv is equal to |pRvqˆ|, so that, by Equation(14.3),

mv “ |pRvqˆ| “ |pFqqˆ| “ q ´ 1 .

If x0, y0 ‰ 0, we have

mv “ Card ta P pRvqˆ : D b P Rv X

1

rRv, ar ´ br2 ´ r P Rvu .

Note that the notation mv coincides with the constant mv, x0, y0 defined before the statementof Theorem 16.1 in both cases.

If Ipx0,y0q is the nonzero fractional ideal

Ipx0,y0q “

! Rv if x0 “ 0 or y0 “ 0 ,Rv X

1rRv X

1r2Rv otherwise,

then

U 18 “!

„

1 b0 1

: b P Ipx0,y0q

)

.

Therefore by Lemma 14.4,

HaarKvppΓ1vq8zKvq “

HaarKvpIpx0,y0qzKvq

rpΓ1vq8 : U 18s“q g´1 NpIpx0,y0qq

mv. (16.8)

251 19/12/2016

Let px0q “ś

p pνppx0q and py0q “

ś

p pνppy0q be the prime decompositions of the principal

ideals px0q and py0q. By the formulas of the prime decompositions of intersections, sums andproducts of ideals in Dedekind rings (see for instance [Nar, §1.1]), we have

px20q X px0y0q X py

20q “ px

20q X py

20q “

ź

p

p2 maxtνppx0q, νppy0qu

andxx0, y0y “

ź

p

pmintνppx0q, νppy0qu .

By the definition of the ideal Ipx0,y0q, by the multiplicativity of the norm, and by Equation(14.4), we hence have if x0 ‰ 0 and y0 ‰ 0

NpIpx0,y0qq pNxx0, y0yq2

|y0|v2 “ N

´

`

px20q X px0y0q X py

20q˘

xx0, y0y2px0q

´2py0q´2¯

“ 1 . (16.9)

If x0 “ 0 or y0 “ 0, thenNpIpx0,y0qq “ NpRvq “ 1 . (16.10)

Lemma 16.3 follows from Equations (16.7), (16.8) and (16.9) if x0 ‰ 0 and y0 ‰ 0 or(16.10) if x0 “ 0 or y0 “ 0. l

Let us state one particular case of Theorem 16.1 in an arithmetic setting, using a congru-ence sugbroup.

Theorem 16.4. Let I be a nonzero ideal of Rv. Then as tÑ `8, we have

pqv2 ´ 1q pqv ` 1q ζKp´1q NpIq

ś

p|Ip1`1

Nppqq

q g´1 qvpqvq

´2tÿ

px,yqPRvˆIxx, yy“Rv , vpyqě´t

∆xy

˚á HaarKv ,

where the product ranges over the prime factors p of the ideal I. Furthermore, if

Ψptq “ Card Rv z

px, yq P Rv ˆ I : xx, yy “ Rv, vpyq ě ´t(

,

then there exists κ ą 0 such that, as tÑ `8,

Ψptq “q 2g´2 qv

pqv2 ´ 1q pqv ` 1q ζKp´1q NpIqś

p|Ip1`1

Nppqqqv

2t `Opqvp2´κqtq .

For every β P s0, 1ln qv

s, there exists κ ą 0 such that for every ψ P C βc pKvq there is an error

term in the above equidistribution claim evaluated on ψ, of the form Ops´κψβq.

Proof. The counting claim is deduced from the equidistribution claim in the same way thatCorollary 16.2 is deduced from Theorem 16.1, noting that the action of Rv by horizontalshears preserves Rv ˆ I.

In order to prove the equidistribution claim, we apply Theorem 16.1 with px0, y0q “ p1, 0qand with G the Hecke congruence subgroup

GI “

"ˆ

a bc d

˙

P GL2pRvq : c P I

*

, (16.11)

252 19/12/2016

which is the preimage of the upper triangular subgroup of GL2pRvIq by reduction moduloI. In this case, the constant mv,x0,y0 appearing in the statement of Theorem 16.1 is equal toq ´ 1 by Equation (16.1). The group GI has finite index in GL2pRvq. The following result iswell-known to arithmetic readers (see for instance [Shi, page 24] when Rv is replaced by Z),we only give a sketch of proof (indicated to us by J.-B. Bost) for the sake of the geometerreaders.

Lemma 16.5. We have

rGL2pRvq : GIs “ NpIqź

p|I

p1`1

Nppqq .

where the product ranges over the prime factors p of the ideal I.

Proof. In this proof, we denote by |E| the cardinality of a finite set E. For every commutativering A with finite group of invertible elements Aˆ, we have

GL2pAq “ď

aPAˆ

ˆ

a 00 1

˙

SL2pAq ,

Hence rGL2pAq : SL2pAqs “ |Aˆ|. Since

ˆ

a 00 1

˙

belongs to GI for all a P pRvqˆ, we have

GI “ď

aPpRvqˆ

ˆ

a 00 1

˙

GI X SL2pRvq ,

so that rGL2pRvq : GIs “ rSL2pRvq : GI X SL2pRvqs.The group morphism of reduction modulo I from SL2pRvq to SL2pRvIq is onto, by an

argument of further reduction to the various prime power factors of I and of lifting elementarymatrices. The order of the upper triangular subgroup of SL2pRvIq is |pRvIqˆ| |RI|, wherepRvIq

ˆ is the group of invertible elements of the ring RvI (that we will see again below).Hence

rGL2pRvq : GIs “ rSL2pRvq : GI X SL2pRvqs “| SL2pRvIq|

|pRvIqˆ| |RI|

“|GL2pRvIq|

|pRvIqˆ|2|RI|. (16.12)

By the multiplicativity of the norm and by the Chinese remainder theorem,3 one reduces theresult to the case when I “ pn is the n-th power of a fixed prime ideal p with norm Nppq “ N ,where n P N. Note that since Rvp is a field, we have

|GL1pRvpq| “ |pRvpqˆ| “ |Rvp| ´ 1 “ N ´ 1

and|GL2pRvpq| “ p|Rvp|

2 ´ 1qp|Rvp|2 ´ |Rvp|q “ N2pN ´ 1q2

`

1`1

N

˘

.

3saying that the rings RvI andś

pRvpvppIq are isomorphic, see for instance [Nar, page 11]

253 19/12/2016

For k “ 1 or k “ 2, the kernel of the morphism of reduction modulo p from GLkpRvIq “GLkpRvp

nq to GLkpRvpq has order Nk2pn´1q. Hence

|GL2pRvIq| “ N4pn´1qN2pN ´ 1q2`

1`1

N

˘

,

and|pRvIq

ˆ| “ Nn´1pN ´ 1q .

Therefore, by Equation (16.12), we have

rGL2pRvq : GIs “N4pn´1qN2pN ´ 1q2

`

1` 1N

˘

N2pn´1qpN ´ 1q2Nn“ Nn

`

1`1

N

˘

.

This proves the result. l

We can now conclude the proof of Theorem 16.4. Note that GL2pRvqp1,0q “ pGIqp1,0q. Theresult then follows from Theorem 16.1 and its Corollary 16.2, using the change of variabless “ pqvq

t, sinceGIp1, 0q “ tpx, yq P Rv ˆ I : xx, yy “ Rvu . l

The following result is a particular case of Theorem 16.4.

Corollary 16.6. Let P0 be a nonzero element of the polynomial ring R “ FqrY s over Fq, andlet P0 “ a0

śki“1pPiq

ni be the prime decomposition of P0. Then as tÑ `8,

pq ` 1qśki“1 q

ni degPip1` q´degPiq

q ´ 1q´2t

ÿ

pP,QqPRˆpP0RqPR`QR“R, degQďt

∆PQ

˚á HaarFqppY ´1qq .

For every β P s0, 1ln qv

s, there exists κ ą 0 such that for every ψ P C βc

`

FqppY ´1qq˘

there isan error term in the above equidistribution claim evaluated on ψ, of the form Ops´κψβq.

Proof. In this statement, we use the standard convention that k “ 0 if P0 is constant,a0 P pFqqˆ and Pi P R is monic.

We apply the first claim of Theorem 16.4 with K “ FqpT q and v “ v8 so that g “ 0,qv “ q and Rv “ R, and with I “ P0R, so that NpIq “

śki“1 q

ni degPi . The result followsfrom Equation (14.5). l

16.2 Mertens’s formula in function fields

In this Section, we recover the function field analogue of Mertens’s classical formula on the av-erage order of the Euler function. We begin with a more general counting and equidistributionresult.

Let m be a (nonzero) fractional ideal of Rv, with norm Npmq. Note that the action ofthe additive group Rv on Kv ˆKv by the horizontal shears k ¨ px, yq “ px` ky, yq preservesmˆm. We consider the counting function ψm : r0,`8r Ñ N defined by

ψmpsq “ Card`

Rv z

px, yq P mˆm : 0 ă Npmq´1Npyq ď s, xx, yy “ m(˘

.

Note that ψm depends only on the ideal class of m and thus we can assume in the computationsthat m is integral.

254 19/12/2016

Corollary 16.7. There exists κ ą 0 such that, as sÑ `8,

ψmpsq “pq ´ 1q q 2g´2 qv

3

pqv2 ´ 1q pqv ` 1q ζKp´1q mv, x0, y0

s2 `Ops2´κq ,

where m “ xx0, y0y. Furthermore, as sÑ `8,

pqv2 ´ 1q pqv ` 1q ζKp´1q mv, x0, y0

pq ´ 1q q g´1 qv3s´2

ÿ

px,yqPmˆmNpmq´1Npyqďs, xx,yy“m

∆xy

˚á HaarKv .

For every β P s0, 1ln qv

s, there exists κ ą 0 such that for every ψ P C βc

`

Kvq˘

there is anerror term in the above equidistribution claim evaluated on ψ, of the form Ops´κψβq.

Theorem 1.14 in the Introduction follows from this result, by taking K “ FqpY q (so thatg “ 0) and v “ v8. In order to simplify the constant, we use Equation (14.5) and the fact thatthe ideal class number of K, that equals the number of orbits of PGL2pFqrY sq on P1pFqpY qq,is 1. Thus, if m “ xx0, y0y then the constant mv, x0, y0 is equal to mv, 1, 0, which is q ´ 1 byEquation (16.1).

Proof. Every nonzero ideal I in Rv is of the form I “ xRv ` yRv for some px, yq P Rv ˆRv ´ tp0, 0qu, see for instance [Nar, page 10]. For all px, yq and pz, wq in Rv ˆ Rv, we havexRv ` y Rv “ z Rv ` wRv if and only if pz, wq P GL2pRvqpx, yq. The ideal class group of Kcorresponds bijectively to the set PGL2pRvq zP1pKq of cusps of the quotient graph of groupsPGL2pRvq zzXv (where Xv is the Bruhat-Tits tree of pPGL2,Kvq ), by the map induced byI “ xRv ` y Rv ÞÑ rx : ys P P1pKq.

Given a fixed ideal m in Rv, we apply Theorem 16.1 with G “ GL2pRvq and px0, y0q P

Rv ˆ Rv ´ tp0, 0qu a fixed pair such that x0Rv ` y0Rv “ m. Using therein the change ofvariable s ÞÑ Npmqs, the result follows from Theorem 16.1 and its Corollary 16.2. l

As already encountered in the proof of Lemma 16.5, the Euler function ϕRv of Rv isdefined on the set of (nonzero, integral) ideals I of Rv by setting4

ϕRvpIq “ CardppRvIqˆq ,

and we denote ϕRvpyq “ ϕRvpy Rvq for every y P Rv. Thus, by the definition of the action ofRv on Rv ˆRv by shears, we have

ψRvpsq “ÿ

yPRv , 0ăNpyqďs

Cardtx P RvyRv : xx, yy “ Rvu

“ÿ

yPRv , 0ăNpyqďs

ϕRvpyq . (16.13)

As a particular application of Corollary 16.7, we get a well-known asymptotic result onthe number of relatively prime polynomials in FqrY s. The Euler function of the ring ofpolynomials R “ FqrY s is then the map φq : R´ t0u Ñ N defined by

φqpQq “ˇ

ˇ

`

RQR˘ˆˇ

ˇ “ Card

P P R : xP,Qy “ R, degP ă degQ(

.

Note that φqpλQq “ φqpQq for every λ P pFqqˆ.4See for example [Ros, §1].

255 19/12/2016

Corollary 16.8 (Mertens’s formula for polynomials). We have

limnÑ`8

1

q2n

ÿ

QPFqrXs, degQďn

φqpQq “pq ´ 1q q

q ` 1.

Proof. We apply the first claim of Corollary 16.7, in the special case when K “ FqpT q andv “ v8 so that g “ 0, qv “ q and Rv “ R, and with m “ Rv, so that mv, x0, y0 “ q ´ 1, inorder to obtain the asymptotic value of ψRvpsq with the change of variable s “ qn. The resultfollows from Equations (16.13) and (14.5). l

The above result is an analog of Mertens’s formula when K is replaced by Q and Rv by Z,see [HaW, Thm. 330]. See also [Grot, Satz 2], [Cos, §4.3], as well as [PaP14b] and [PaP16b,§5] for further developments.

A much more precise result than Corollary 16.8 can be obtained by purely number the-oretical means as follows. The average value of φq is computed in [Ros, Prop. 2.7]: Forn ě 1,

ÿ

deg f“n, f monic

φqpfq “ q2np1´1

qq .

This givesř

deg f“n φqpfq “ q2n pq´1q2

q , so that

ÿ

0ădeg fďn

φqpfq “nÿ

k“1

q2k pq ´ 1q2

q“ qpq ´ 1q2

q2n ´ 1

q2 ´ 1“qpq ´ 1qpq2n ´ 1q

pq ` 1q,

from which Corollary 16.8 easily follows.

256 19/12/2016

Chapter 17

Equidistribution and counting ofquadratic irrational points innon-Archimedean local fields

Let Kv be a non-Archimedean local field, with valuation v, valuation ring Ov, choice of uni-formiser πv, and residual field kv of order qv. Let Xv be the Bruhat-Tits tree of pPGL2,Kvq.1

In this Chapter, we give counting and equidistribution results in Kv “ B8Xv´t8u of an orbitunder a lattice of PGL2pKvq of a fixed point of a loxodromic element of this lattice. We usethese results to deduce equidistribution and counting results of quadratic irrational elementsin non-Archimedean local fields.

When Xv is replaced by a real hyperbolic space, or by a more general simply connectedcomplete Riemannian manifold with negative sectional curvature, there are numerous quan-titative results on the density of such an orbit, see the works of Patterson, Sullivan, Hill,Velani, Stratmann, Hersonsky-Paulin, Parkkonen-Paulin. See for instance [PaP16a] for ref-erences. The arithmetic applications when Xv is replaced by the upper halfspace model ofthe real hyperbolic space of dimension 2, 3 or 5 are counting and equidistribution resultsof quadratic irrational elements in R, C and the Hamiltonian quaternions. See for instance[PaP12, Coro. 3.10] and [PaP14b].

17.1 Counting and equidistribution of loxodromic fixed points

An element γ P PGL2pKvq is said to be loxodromic if it is loxodromic on the (geometricrealisation of the) simplicial tree Xv.2 Its translation length is

λpγq “ minxPV Xv

dpx, γxq ą 0 ,

and the subsetAxγ “ tx P V Xv : dpx, γxq “ λpγqu

is the image of a (discrete) geodesic line in Xv, which we call the (discrete) translation axis ofγ. The points at infinity of Axγ are denoted by γ´ and γ` chosen so that γ translates away

1See Section 15.1.2See Section 2.2.

257 19/12/2016

from γ´ and towards γ` on Axγ . Note that for every γ1 P PGL2pKvq, we have

γ1Axγ “ Axγ1γ pγ1q´1 and γ1 γ˘ “ pγ1γ pγ1q´1q˘ .

If Γ is a discrete subgroup of PGL2pKvq and if α is one of the two fixed points of aloxodromic element of Γ, we denote the other fixed point of this element by ασ. Since Γis discrete, the translation axes of two loxodromic elements of Γ coincide if they have acommon point at infinity. Hence ασ is uniquely defined. We define the complexity hpαq of theloxodromic fixed point α by

hpαq “1

|α´ ασ|v(17.1)

if α, ασ ‰ 8, and by hpαq “ 0 if α or ασ is equal to 8. We define ια “ 2 if there exists anelement γ P Γ such that γ ¨ α “ ασ, and ια “ 1 otherwise.3

Following [Ser3, II.1.2], we denote by PGL2pKvq` the kernel of the group morphism

PGL2pKvq Ñ Z2Z defined by γ “ rgs ÞÑ vpdet gq mod 2. The definition does not dependon the choice of a representative g P GL2pKvq of an element γ P PGL2pKvq, since

v`

det

ˆ

λ 00 λ

˙

˘

“ 2 vpλq

is even for every λ P pKvqˆ. Note that when Kv is as in Section 14.2 the completion of a

function field over Fq endowed with a valuation v, with associated affine function ring Rv, thegroup Γv “ PGL2pRvq is contained in PGL2pKvq

`, see [Ser3, II.1.2]: For every g P GL2pRvq,since det g P pRvq

ˆ “ pFqqˆ,4 we have vpdet gq “ 0.

The following result proves the equidistribution in Kv of the loxodromic fixed pointswith complexity at most s in a given orbit by homographies under a lattice in PGL2pKvq ass Ñ `8, and its associated counting result. If ξ P B8Xv “ P1pKvq and Γ is a subgroup ofPGL2pKvq, we denote by Γξ the stabiliser in Γ of ξ.

Theorem 17.1. Let Γ be a lattice in PGL2pKvq`, and let γ0 P Γ be a loxodromic element of

Γ. Then as sÑ `8,

pqv ` 1q2 VolpΓzzXvq2 q2

v VolpΓγ´0zzAxγ0q

s´1ÿ

α PΓ¨γ´0 , hpαqďs

∆α˚á HaarKv

and

Cardtα P pΓ ¨ γ´0 q X Ov : hpαq ď su „2 q2

v VolpΓγ´0zzAxγ0q

pqv ` 1q2 VolpΓzzXvqs .

When Γ is geometrically finite, there is an error term of the form Ops1´κq for some κ “κΓ ą 0 in the counting claim and, for every β P s0, 1

ln qvs, an error term of the form Ops´κψβq

for some κ ą 0 in the equidistribution claim evaluated on any β-Hölder-continuous functionψ : Ov Ñ C.

Proof. The second result follows from the first one by integrating on the characteristicfunction of the compact-open subset Ov, whose Haar measure is 1.

3The groups GL2pKvq and PGL2pKvq act on P1pKvq “ Kv Y t8u by homographies. See Section 15.1.

4See Equation (14.3).

258 19/12/2016

In order to prove the equidistribution result, we apply Theorem 15.4 with D´ :“ t˚vu

and D` :“ Axγ0 . The families D˘ “ pγD˘qγPΓΓD˘are locally finite, since Γ is discrete and

the stabiliser ΓD` of D` acts cocompactly on D`. Furthermore, σ´D` is finite and nonzeroby Equation (8.11). Since Γ is contained in PGL2pKvq

`, the length spectrum LΓ of Γ iscontained in 2Z by [Ser3, II.1.2, Coro.]. Hence, it is equal to 2Z by the equivalence given byEquation (4.13).

For every γ P Γ such that dpD´, γD`q ą 0,5 let ργ be the geodesic ray starting at time0 from the origin of α´e, γ (which is ˚v) with point at infinity γ ¨ γ´0 . Since Xv is a tree andγ ¨ γ´0 is one of the two endpoints of γD`, the geodesic segment α´e, γ |r0,dpD´,γD`qs is an initialsubsegment of ργ .6 Therefore, by Equation (15.15), for the weak-star convergence of measureson B1

`D´, we have

limnÑ`8

pqv2 ´ 1qpqv ` 1q

2 q3v

VolpΓzzXvqσ´D`

qv´n

ÿ

γPΓΓD`0ădpD´, γD`qďn

∆γ¨γ´0“ pB`q˚rσ

`

D´ . (17.2)

Furthermore, for every β P s0, 1ln qv

s, by the comment following Equation (15.15), we have anerror term of the form Opψβ ln qv e

´κnq for some κ ą 0 in the above formula when evaluatedon ψ P C β ln qv

c pB8Xvq, where B8Xv is endowed with the visual distance d˚v . Note that onx, y P Ov, the visual distance d˚v and the distance px, yq ÞÑ |x´ y|v are related by

|x´ y|v “ dH8px, yqln qv “ d˚vpx, yq

ln qv ,

see Equation (15.3). Hence we have an error term Opψβ e´κnq for some κ ą 0 in the above

formula when evaluated on ψ P C βc pOvq, where Ov is endowed with the distance px, yq ÞÑ

|x´ y|v.

8

˚v

H8

Kv “ B8Xv ´ t8u

kk

BH8

Ov π´kv ` Ov

pH8pπ´kv q

γ ¨ γ´0γ ¨ γ`0

γD`

Let us fix for the moment k P N. For every ξ P π´kv ` Ov, we have |ξ|v “ q´vpξqv “ qkv if

k ě 1 and |ξ|v ď 1 if k “ 0. By restricting the measures to the compact-open subset π´kv `Ov

5that is, such that ˚v R γD`6It connects ˚v to its closest point PγD`p˚vq on γD`.

259 19/12/2016

and by Proposition 15.2 (1), we have, with the appropriate error term when Γ is geometricallyfinite and k “ 0,

limnÑ`8

pqv2 ´ 1qpqv ` 1q

2 q3v

VolpΓzzXvqσ´D`

qv´n

ÿ

γPΓΓD`

γ¨γ´0 Pπ´kv `Ov

0ădpD´, γD`qďn

∆γ¨γ´0“ q´2 k

v HaarKv |pπvq´k`Ov.

(17.3)If β P Γ is loxodromic and satisfies β´ P π´kv `Ov and β` R π´kv `Ov, then the translation

axis of β passes at distance at most 2k from ˚v, since it passes through PH8pπ´kv q which is

the closest point on H8 to π´kv . If β P Γ is loxodromic and satisfies β´, β` P π´kv `Ov, then

dp˚v,Axβq “ 2k ` dpH8,Axβq .

Furthermore, we have, by Equations (15.2) and (2.8)

|β´ ´ β`|v “ dH8pβ´, β`qln qv “ q

´dpH8,Axβqv .

Therefore by the definition of the complexity in Equation (17.1), we have for these elements

hpβ´q “1

|β´ ´ β`|v“ q

dpH8,Axβqv “ q

dp˚v ,Axβq´2kv . (17.4)

Since the family D` “ pγD`qγPΓΓD`is locally finite, there are only finitely many elements

γ P ΓΓD` such that γD` “ Axγγ0γ´1 is at distance at most 2k from ˚v. Hence for all butfinitely many γ P ΓΓD` such that γ ¨ γ´0 “ pγγ0γ

´1q´ P π´kv ` Ov, we have γ ¨ γ`0 “

pγγ0γ´1q` P π´kv ` Ov and, using Equation (17.4) with β “ γγ0γ

´1,

hpγ ¨ γ´0 q “ q dpD´,γD`q´2k

v .

Therefore, using the change of variable s “ q n´2kv , Equation (17.3) becomes

limsÑ`8

pqv2 ´ 1qpqv ` 1q

2 q3v

VolpΓzzXvqσ´D`

s´1ÿ

γPΓΓD`

γ¨γ´0 Pπ´kv `Ov

0ăhpγ¨γ´0 qďs

∆γ¨γ´0“ HaarKv |pπvq´k`Ov

. (17.5)

Note that the stabiliser Γγ´0of γ´0 in Γ has index ιγ´0 in ΓD` by the definition of ιγ´0 and

that ΓΓγ´0identifies with Γ ¨ γ´0 by the map γΓγ´0

ÞÑ γ ¨ γ´0 . Since`

pπvq´k ` Ov

˘

kPN is acountable family of pairwise disjoint compact-open subsets covering Kv, and since the supportof any continuous function with compact support is contained in finitely many elements ofthis family, we have

limsÑ`8

pqv2 ´ 1qpqv ` 1q

2 q3v ιγ´0

VolpΓzzXvqσ´D`

s´1ÿ

αPΓ¨γ´00ăhpαqďs

∆α “ HaarKv , (17.6)

with the appropriate error term when Γ is geometrically finite.

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Recall that by Equation (8.11), if the Patterson measures are normalised to be probabilitymeasures, then

σ´D` “qv ´ 1

qv ` 1VolpΓD`zzD`q .

Hence if instead the Patterson densities are normalised to have total mass qv`1qv

as in Propo-sition 15.2 (2), then

σ´D` “qv ´ 1

qvVolpΓD`zzD`q .

Note that, since ιγ´0 “ rΓAxγ0: Γγ´0

s,

VolpΓγ´0zzAxγ0q “ ιγ´0

VolpΓAxγ0zzAxγ0q .

Equation (17.6) thus gives the equidistribution result in Theorem 17.1. l

In the following two Sections, we use Theorem 17.1 to deduce counting and equidistri-bution results of elements of non-Archimedean local fields that are quadratic irrational overappropriate subfields, when an appropriate algebraic complexity tends to infinity.

17.2 Counting and equidistribution of quadratic irrationals inpositive characteristic

Let K be a function field over Fq, let v be a (normalised discrete) valuation of K, and let Rvbe the affine function ring associated with v. We assume in this Section that the characteristicof K is different from 2.7

An element β P Kv is quadratic irrational over K if β R K and β is a root of a quadraticpolynomial aβ2 ` bβ ` c for some a, b, c P K with a ‰ 0. The Galois conjugate βσ of β isthe other root of the same polynomial. Let trpβq “ β ` βσ and npβq “ ββσ be the relativetrace and relative norm of β. It is easy to check that βσ ‰ β since the characteristic of K isdifferent from 2. The next proposition gives a characterisation of quadratic irrationals overK.

Proposition 17.2. Let β P Kv. The following assertions are equivalent:

(1) β is quadratic irrational over K,

(2) β is a fixed point of a loxodromic element of PGL2pRvq.

Proof. The fact that (2) implies (1) is immediate since PGL2pRvq acts by homographies.The converse is classical, see for instance [PaP11b, Lem. 6.2] in the Archimedean case and[BerN] above its Section 5 when K “ FqpY q and v “ v8. l

If β P Kv is quadratic irrational over K, its Galois conjugate βσ is the other fixed pointof a loxodromic element of PGL2pRvq fixing β, hence the notations βσ in this Section and inSection 17.1 coincide.

The actions by homographies of the groups GL2pRvq and PGL2pRvq on KvYt8u preservethe set of quadratic irrationals over K. Contrary to the case of rational points, both groupsact with infinitely many orbits.

7This is equivalent to q being odd.

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The complexity of a quadratic irrational α P Kv over K is

hpαq “1

|α´ ασ|v,

see for instance [HeP4, §6] for motivations and results when K “ FqpY q and v “ v8. Notethat this complexity is invariant under the action of the stabiliser GL2pRvq8 of8 in GL2pRvq,which is its upper triangular subgroup. In particular, it is invariant under the action of Rv bytranslations.8 In [PaP12], where K and | ¨ |v are replaced by Q and its Archimedean absolutevalue, there was, for convenience, an extra factor 2 in the numerator of the complexity, whichis not needed here. We refer for instance to [PaP12, Lem. 4.2] for the connection of thiscomplexity to the standard height, and to [PaP12, §4.2, 4.4] and [PaP12, §6.1] for studiesusing this complexity.

The complexity hp¨q satisfies the following elementary properties, giving in particular itsbehaviour under the action of PGL2pRvq by homographies on the quadratic irrationals in Kv

over K. We also give the well-known computation of the Jacobian of the Haar measure for thechange of variables given by homographies, and prove (using the properties of the complexityhp¨q) the invariance of a measure which will be useful in Section 18.1.

For all g “ˆ

a bc d

˙

P GL2pKvq and z P Kv such that g ¨ z ‰ 8, let

jgpzq “|det g|v|c z ` d| 2

v

.

Proposition 17.3. Let α P Kv be a quadratic irrational over K.

(1) We have hpαq “1

a

| trpαq2 ´ 4 npαq|v.

(2) For every g “ˆ

a bc d

˙

P GL2pKq with | det g|v “ 1, we have

hpg ¨ αq “ | npd` c αq|v hpαq .

(3) If Qα : RvˆRv Ñ r0,`8r is the map px, yq ÞÑ | npx´y αq|v, then for every g P GL2pRvq,we have

Qg¨α “hpαq

hpg ¨ αqQα ˝ g

´1 .

In particular, if g P GL2pRvq fixes α, then

Qα ˝ g “ Qα .

(4) For all x, y, z P Kv and g P GL2pKvq such that g ¨ x, g ¨ y, g ¨ z ‰ 8, we have

|g ¨ x´ g ¨ y| 2v “ |x´ y| 2

v jgpxq jgpyq

and

jgpzq “dpg´1q˚HaarKv

dHaarKvpzq .

8This is a particular case of Proposition 17.3 (2) below.

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(5) The measure

dµpzq “dHaarKvpzq

|z ´ α|v |z ´ ασ|v

on Kv ´ tα, ασu is invariant under the stabiliser of α in PGL2pRvq

Proof. (1) This follows from the formula pα´ ασq2 “ pα` ασq2 ´ 4αασ.

(2) Since g has rational coefficients (that is, coefficients in K), we have

g ¨ α´ pg ¨ αqσ “ g ¨ α´ g ¨ ασ “aα` b

cα` d´aασ ` b

cασ ` d

“pad´ bcqpα´ ασq

pcα` dqpcασ ` dq“pdet gqpα´ ασq

npd` cαq.

Taking absolute values and inverses, this gives Assertion (2).

(3) Let g “ˆ

a bc d

˙

P GL2pRvq. Note that g´1 ¨ α “ dα´ba´cα . For all x, y P Rv, we hence have

n`

pax` byq ´ pcx` dyqα˘

“ n`

xpa´ cαq ´ ypdα´ bq˘

“ n`

xpa´ cαq ´ ypa´ cαq g´1 ¨ α˘

“ npa´ cαq npx´ y g´1 ¨ αq .

Taking absolute values and using Assertion (2), we have

Qα ˝ g “hpg´1 ¨ αq

hpαqQg´1¨α .

Assertion (3) follows by replacing g by its inverse.

(4) Let g “ˆ

a bc d

˙

P GL2pKvq. As seen in the proof of Assertion (2), we have

g ¨ x´ g ¨ y “pdet gqpx´ yq

pc x` dqpc y ` dq.

Taking absolute values and squares, this gives the first claim of Assertion (4).Recall that a homography z ÞÑ a z`b

c z`d is holomorphic9 on Kv ´ t´dc u, with derivative

z ÞÑ ad´bcpcz`dq2

. Hence infinitesimally close to z, the homography acts (up to translations whichleave the Haar measure invariant) by a homothety of ratio ad´bc

pcz`dq2. By Equation (14.6), this

proves that

dHaarKvpg ¨ zq “| det g|v|c z ` d| 2

v

dHaarKvpzq ,

as wanted.

9We refer for instance to [Ser4] for background on holomorphic functions on non-Archimedean local fields.

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(5) Let g “ˆ

a bc d

˙

P GL2pRvq fixing α. Note that an element of GL2pRvq which fixes α also

fixes ασ. By Assertion (4), we have

dµpg ¨ zq “dHaarKvpg ¨ zq

|g ¨ z ´ α|v |g ¨ z ´ ασ|v“

dHaarKvpg ¨ zq

|g ¨ z ´ g ¨ α|v |g ¨ z ´ g ¨ ασ|v

“jgpzq dHaarKvpzq

|z ´ α|va

jgpzq jgpαq |z ´ ασ|va

jgpzq jgpασq

“1

a

jgpαq jgpασqdµpzq .

Again by Assertion (4), since | det g|v “ 1 as det g P pRvqˆ “ pFqqˆ by Equation (14.3), by

Assertion (2) and since g fixes α, we have

1a

jgpαq jgpασq“ |c α` d|v |c α

σ ` d|v “ | npc α` dq|v

“hpg ¨ αq

hpαq“ 1 .

The result follows. l

Let G be a finite index subgroup of GL2pRvq. We say that a quadratic irrational β P Kv

over K is G-reciprocal (simply reciprocal if G “ GL2pRvq) if some element of G maps β to βσ.We define the G-reciprocity index ιGpβq as 2 if β is G-reciprocal and 1 otherwise. Similarly,we say that a loxodromic element γ of G is G-reciprocal (simply reciprocal if G “ GL2pRvq)if there exists an element in G that switches the two fixed points of γ.

Proposition 17.4. Let G be a finite index subgroup of GL2pRvq, and let γ be a loxodromicelement of G. The following assertions are equivalent:

(1) γ is conjugate in G to γ1γ´1 for some γ1 P G pointwise fixing Axγ,

(2) the loxodromic element γ is G-reciprocal,

(3) the quadratic irrational γ´ is G-reciprocal.

When G “ GL2pRvq, Assertions (1), (2) and (3) are also equivalent to

(4) the image of γ2γ in PGL2pRvq, for some γ2 P G pointwise fixing Axγ, is conjugate tothe image in PGL2pRvq of tγ.

Proof. Most of the proofs are similar to the ones when Rv, K and v are replaced by Z, Qand its Archimedean absolute value, see for instance [PaP12]. We only give hints for the sakeof completeness. Let α “ γ´.

If α is G-reciprocal, then let β P G be such that β ¨ α “ ασ. Since Rv Ă K, we haveβ ¨ ασ “ α. Hence βγβ´1 is a loxodromic element of G fixing α and ασ, having the sametranslation length as γ, but translating in the opposite direction on Axγ . Hence γ1 “ βγβ´1γfixes pointwise Axγ . Therefore (3) implies (1).

If β P G conjugates γ to γ1γ´1 for some γ1 P G pointwise fixing Axγ , then β preserves theset tα, ασu. Hence, it preserves the translation axis of γ but it switches α and ασ since γ andγ1γ´1 translate in opposite directions on Axγ . Therefore (1) implies (2).

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The fact that (2) implies (3) is immediate, since ασ “ γ`.

The equivalence between (1) and (4) when G “ GL2pRvq follows from the fact that thestabiliser of Axγ normalises the pointwise stabiliser of Axγ , and from the formula

t

ˆ

a bc d

˙´1

“1

ad´ bc

ˆ

0 1´1 0

˙ˆ

a bc d

˙ˆ

0 1´1 0

˙´1

which is valid over any field. l

The following result says that any orbit of a given quadratic irrational in Kv over K, byhomographies under a given finite index subgroup of the modular group PGL2pRvq, equidis-tributes to the Haar measure on Kv. Again, note that we are not assuming the finite indexsubgroup to be a congruence subgroup.

Theorem 17.5. Let G be a finite index subgroup of GL2pRvq. Let α0 P Kv be a quadraticirrational over K. Then, as sÑ `8,

pqv ` 1q2 ζKp´1q m0 rGL2pRvq : Gs

2 q2v pq ´ 1q |vptr g0q|

s´1ÿ

αPG¨α0 : hpαqďs

∆α˚á HaarKv ,

where g0 P G fixes α0 with vptr g0q ‰ 0, and where m0 is the index of gZ0 in the stabiliser ofα0 in G. Furthermore, there exists κ ą 0 such that, as sÑ `8,

Cardtα P pG ¨ α0q X Ov : hpαq ď su “2 q2

v pq ´ 1q |vptr g0q|

pqv ` 1q2 ζKp´1q m0 rGL2pRvq : Gss`Ops1´κq .

For every β P s0, 1ln qv

s, there exists κ ą 0 such that for every ψ P C βc

`

Kvq˘

there is anerror term in the above equidistribution claim evaluated on ψ, of the form Ops´κψβq.

Proof. We apply Theorem 17.1 with Γ the image of G in PGL2pRvq and with γ0 the imagein PGL2pRvq of the element g0 introduced in the statement. Note that Γ, which is containedin Γv, is indeed contained in PGL2pKvq

`.By Equation (15.6), the translation length of g in Xv is 2 |vptr gq|, and g P GL2pRvq is

loxodromic if and only if vptr gq ‰ 0. This implies that g0 exists, since G has finite index inGL2pRvq, and such an element exists in GL2pRvq by Proposition 17.2. Furthermore

λpγ0q “ 2 |vptr g0q| .

Since the centre of GL2pKvq acts trivially by homographies, we have

G ¨ α0 “ Γ ¨ α0 .

For every α P G ¨ α0, the complexities hpαq, when α is considered as a quadratic irrational orwhen α is considered as a loxodromic fixed point, coincide.

Since the centre ZpGq of G acts trivially by homographies, by the definition of m0 in thestatement, we have

rΓγ´0: γZ0 s “

rGα0 : gZ0 s

|ZpGq|“

m0

|ZpGq|.

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Therefore,

VolpΓγ´0zzAxγ0q “

1

rΓγ´0: γZ0 s

VolpγZ0 zzAxγ0q “λpγ0q

rΓγ´0: γZ0 s

“2 |vptr g0q| |ZpGq|

m0. (17.7)

Theorem 17.5 now follows from Theorem 17.1 using Equations (16.6) and (17.7). l

Example 17.6. (1) Theorem 1.13 in the Introduction follows from this result, by takingK “ FqpY q and v “ v8, and by using Equation (14.5) in order to simplify the constant.

(2) Let GI be the Hecke congruence subgroup associated with a nonzero ideal I of Rv, seeEquation (16.11). By Lemma 16.5, we have, as sÑ `8,

pqv ` 1q2 ζKp´1q m0 NpIqś

p|Ip1`1

Nppqq

q2v pq ´ 1q |vptr g0q|

s´1ÿ

αPGI ¨α0 : hpαqďs

∆α˚á HaarKv .

We conclude this Section by a characterisation of quadratic irrationals and reciprocalquadratic irrationals in the field of formal Laurent series FqppY ´1qq in terms of continuedfractions. When FqrY s, FqpY q and v8 are replaced by Z, Q and its Archimedean absolutevalue, we refer for instance to [Sarn] and [PaP12, Prop. 4.3] for characterisations of reciprocalquadratic irrationals.

Recall that Artin’s continued fraction expansion of f P FqppY ´1qq´FqpY q is the sequencepai “ aipfqqiPN in FqrY s with deg ai ą 0 if i ą 0 such that

f “ a0 `1

a1 `1

a2 `1

a3 `1

. . .

.

See for instance the surveys [Las, Sch2], and [Pau1] for a geometric interpretation. We saythat the continued fraction expansion of f is eventually periodic if there exist n P N andN P N´ t0u such that an`i “ an`N`i for every i P N, and we write

f “ ra0, . . . , an´1, an, . . . , an`N´1s .

Such a sequence an, . . . , an`N´1 is called a period of f , and if of minimal length, it is welldefined up to cyclic permutation.

Two elements β, β1 P FqppY ´1qq are in the same PGL2pFqrY sq-orbit if and only if theircontinued fraction expansions have equal tails up to an invertible element of FqrY s by [BerN,Theo. 1]. More precisely, β, β1 P Kv are in the same PGL2pFqrY sq-orbit if and only if thereexist m,n P N and x P Fˆq such that for every k P N, we have an`kpβ1q “ xp´1qk am`kpβq.

Proposition 17.7. Let K “ FqpY q and v “ v8.

(1) An element α P Kv ´ K is quadratic irrational over K if and only if its continuedfraction expansion of β is eventually periodic, and if and only if it is a fixed point of aloxodromic element of PGL2pFqrY sq.

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(2) A quadratic irrational α P Kv is reciprocal if and only if the period a0, . . . , aN´1 of thecontinued fraction expansion of α is palindromic up to cyclic permutation and invertibleelements, in the sense that there exist x P Fˆq and p P N such that for k “ 0, . . . , N ´ 1,we have ak`p “ xp´1qkaN´k´1 (with indices modulo N).

Proof. (1) The equivalence of being quadratic irrational and having an eventually periodiccontinued fraction expansion is well-known, see for instance the survey [Las, Theo. 3.1]. Therest of the claim follows from Proposition 17.2.

(2) The proof is similar to the Archimedean case in [Per, §23].10 For every quadratic irrationalf P FqppY ´1qq, up to the action of GL2pFqrY sq, we may assume that f, pfσq´1 P Y ´1FqrrY ´1ss

and f “ r0, a1, a2, . . . , ans. Then we may define by induction quadratic irrationals f2, . . . , fn PFqppY ´1qq over FqpY q such that

1

f“ a1 ` f2,

1

f2“ a2 ` f3, . . . ,

1

fn`1“ an`1 ` fn,

1

fn“ an ` f .

Passing to the Galois conjugates, we have

1

fσ“ a1 ` f

σ2 ,

1

fσ2“ a2 ` f

σ3 , . . . ,

1

fσn“ an ` f

σ .

Taking these equations in the reverse order, we have

1

´ 1fσ“ an ´

1

fσn,

1

´ 1fσn

“ an´1 ´1

fσn´1

, . . . ,1

´ 1fσ2

“ a1 ´1

fσ,

so that, since ´ 1fσ P Y

´1FqrrY ´1ss, we have

´1

fσ“ r0, an, . . . , a2, a1s .

Therefore fσ “ r´an, . . . ,´a2,´a1s. Thus, if f and fσ are in the same orbit, the periods arepalindromic by [BerN, Theo. 1]. l

17.3 Counting and equidistribution of quadratic irrationals inQp

There are interesting arithmetic (uniform) lattices of PGL2pQpq constructed using quaternionalgebras. In this Section, we study equidistribution properties of loxodromic fixed pointselements of these lattices. We use [Vig] as our standard reference on quaternion algebras.

Let F be a field and let a, b P Fˆ. Let D “ pa,bF q be the quaternion algebra over F

with basis 1, i, j, k as a F -vector space such that i2 “ a, j2 “ b and ij “ ji “ ´k. Ifx “ x0 ` x1i` x2j ` x3k P D, then its conjugate is

x “ x0 ´ x1i´ x2j ´ x3k ,

10See also [BerN, Coro. 1] by relating, using twice the period, what the authors call the ´ continued fractionexpansion to the standard expansion.

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its (reduced) norm isNpxq “ xx “ x2

0 ´ a x21 ´ b x

22 ` ab x

23

and its (reduced) trace isTrpxq “ x` x “ 2x0 .

Let us fix two negative rational integers a, b and let D “ pa,bQ q . For every field extension Eof Q, we denote by DE the quaternion algebra DbQE over E, and we say that D splits overE if the E-algebra D bQ E is isomorphic to M2pEq. The assumption that a, b are negativeimplies that D does not split over R. Furthermore, when p P N is an odd prime, D splits overQp if and only if the equation a x2 ` b y2 “ 1 has a solution in Qp, see [Vig, page 32].

The reduced discriminant of D is

DiscD “ź

qPRampDq

q .

where RampDq is the finite set of primes p such that D does not split over Qp.For instance, the quaternion algebra D “ p´1,´1

Q q splits over Qp if and only if p ‰ 2, henceit has reduced discriminant 2.

Assume from now on that p P N is a positive rational prime such that D splits over Qp

and, for simplicity, that Qp contains square roots?a and

?b of a and b. For example, if

a “ b “ ´1, this is satisfied if p ” 1 mod 4. We then have an isomorphism of Qp-algebrasθ “ θa, b : DQp ÑM2pQpq defined by

θpx0 ` x1i` x2j ` x3kq “

˜

x0 ` x1?a

?b px2 `

?a x3q

?b px2 ´

?a x3q x0 ´ x1

?a

¸

, (17.8)

so thatdetpθpxqq “ Npxq and trpθpxqq “ Trpxq .

If the assumption on the existence of the square roots in Qp is not satisfied, we can replaceQp by an appropriate finite extension, and prove equidistribution results in this extension.

Let O be a Z“

1p

‰

-order in DQp , that is, a finitely generated Z“

1p

‰

-submodule of DQp gener-ating DQp as a Qp-vector space, which is a subring of DQp . Let O1 be the group of elementsof norm 1 in O. Then the image Γ1

O of θpO1q in PGL2pQpq is a cocompact lattice, see forinstance [Vig, Sect. IV.1]. In fact, this lattice is contained in PSL2pQpq, hence in PGL2pQpq

`.We denote by Xp the Bruhat-Tits tree of pPSL2,Qpq, which is pp` 1q-regular.

The next result computes the covolume of this lattice.11

Proposition 17.8. Let D be a quaternion algebra over Q which splits over Qp and does notsplit over R. If Omax is a maximal Z

“

1p

‰

-order in DQp containing O, then

VolpΓ1OzzXpq “ rO

1max : O1s

p

12

ź

q|DiscD

pq ´ 1q .

Proof. We refer to [Vig, page 53] for the (common) definition of the discriminant DiscpQpq

of the local field Qp and DiscpDQpq of the quaternion algebra DQp over the local field Qp. Wewill only use the facts that DiscpQpq “ 1 as it easily follows from the definition, and that

DiscpDQpq “ DiscpQpq4NppZpq2 “ p2 (17.9)

11The index q ranges over the primes dividing DiscD, that is, over the elements of RampDq .

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which follows by [Vig, Lem. 4.7, page 53] and [Vig, Cor. 1.7, page 35] for the first equalityand NppZpq “ CardpZppZpq “ CardpZpZq “ p for the second one.

We refer to [Vig, Sect. II.4, page 52] for the definition of the Tamagawa measure µT onXˆ when X “ DQp or X “ Qp. It is a Haar measure of the multiplicative locally compactgroup Xˆ, and understanding its explicit normalisation is the main point of this proposition.By [Vig, Lem. 4.6, page 52],12 with dx the Haar measure on the additive group X,13 with xthe module of the left multiplication by x P Xˆ on the additive group X,14 we have

dµTpxq “1

a

DiscpXq xdx .

By [Vig, Lem. 4.3, page 49], identifying DQp to M2pQpq by θ, the measure of GL2pZpq for themeasure 1

p1´p´1q xdx is 1´ p´2. Hence, by scaling and by Equation (17.9), we have

µTpGL2pZpqq “p1´ p´2qp1´ p´1q

a

DiscpDQpq“pp2 ´ 1qpp´ 1q

p4.

Again by [Vig, Lem. 4.3, page 49], the mass of Zˆp for the measure 1p1´p´1q x

dx on Qˆp is 1,hence by scaling

µTpZˆp q “1´ p´1

a

DiscpQpq“p´ 1

p.

By [Vig, page 54], since we have an exact sequence

1 ÝÑ SL2pQpq ÝÑ GL2pQpqdetÝÑ Qˆp ÝÑ 1 ,

the Tamagawa measure of GL2pQpq disintegrates by the determinant over the Tamagawameasure of Qˆp with conditional measures the translates of a measure on SL2pQpq, called theTamagawa measure of SL2pQpq and again denoted by µT. Thus,

µTpSL2pZpqq “µTpGL2pZpqqµTpZˆp q

“p2 ´ 1

p3

By Example 3 on page 108 of [Vig], since the Z“

1p

‰

-order Omax is maximal, we have, withG “ θpO1

maxq,

µTpGzSL2pQpqq “1

24p1´ p´2q

ź

q|DiscD

pq ´ 1q .

Since GL2pQpq acts transitively on V Xp with stabiliser of the base point ˚ “ rZpˆZps themaximal compact subgroup GL2pZpq,15 and by the centred equation mid-page 116 of [Ser3],we have

VolpGzzXpq “ÿ

rxsPGzV Xp

1

|Gx|“µTpGzGL2pQpqq

µTpGL2pZpqq“µTpGzSL2pQpqq

µTpSL2pZpqq

“p

24

ź

q|DiscD

pq ´ 1q .

12See also the top of page 55 in loc. cit.13with a normalisation that does not need to be made precise14so that pMxq˚dx “ x dx where Mx : y ÞÑ xy is the left multiplication by x on X15See Section 15.1.

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The natural homomorphism G “ θpO1maxq Ñ Γ1

Omaxis 2-to-1 and rΓ1

Omax: Γ1

O s “ rO1max : O1s,

so thatVolpΓ1

OmaxzzXpq “ 2 VolpGzzXpq .

Proposition 17.8 follows. l

Note that the fixed points z for the action on P1pQpq “ QpYt8u by homographies of theelements in the image of θpDq are quadratic over Qp

?a,?bq. More precisely, z?

bis quadratic

over Qp?aq. An immediate application of Theorem 17.1, using Proposition 17.8, gives the

following result of equidistribution of quadratic elements in Qp over Qp?a,?bq.

Theorem 17.9. Let Γ be a finite index subgroup of Γ1O , and let γ0 P Γ be a loxodromic element

of Γ. Then as sÑ `8,

pp` 1q2ś

q|DiscDpq ´ 1q rO1

max : O1s rΓ1O : Γs

24 p VolpΓγ´0zzAxγ0q

s´1ÿ

α PΓ¨γ´0 , hpαqďs

∆α˚á HaarQp ,

where Omax is a maximal Z“

1p

‰

-order in DQp containing O, and there exists κ ą 0 such thatas sÑ `8

Cardtα P pΓ ¨ γ´0 q X Zp : hpαq ď su

“24 p VolpΓγ´0

zzAxγ0q

pp` 1q2ś

q|DiscDpq ´ 1q rO1

max : O1s rΓ1O : Γs

s`Ops1´κq . l

Assume furthermore that the positive rational prime p P N is such that p ” 1 mod 4

and that the integer p2´14 is not of the form 4ap8b ` 7q for a, b P N (for instance p “ 5). By

Legendre’s three squares theorem (see for instance [Gros]), there exist x11, x12, x13 P Z such thatp2´1

4 “ x112` x12

2` x13

2. Hence there are x1, x2, x3 P 2Z such that p2 ´ 1 “ x12 ` x2

2 ` x32.

A standard consequence of Hensel’s theorem says that when p is odd, a number n P Zhas a square root in Zp if n is relatively prime to p and has a square root modulo p, see forinstance [Kna, page 351]. Thus, 1´ p2 has a square root in Zp, that we denote by

a

1´ p2.As noticed above, since p ” 1 mod 4, the element ´1 has a square root in Qp, that we denoteby ε. The element

α0 “ε x1 `

a

1´ p2

x3 ` ε x2

is a quadratic irrational in Qp over Qpεq.

The following result is a counting and equidistribution result of quadratic irrationals overQpεq in Qp. We denote by ασ the Galois conjugate of a quadratic irrational α in Qp overQpεq, and by

hpαq “1

|α´ ασ|p

the complexity of α.

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Theorem 17.10. Let D “ p´1,´1

Q q be Hamilton’s quaternion algebra over Q. Let p P N be

a positive rational prime with p ” 1 mod 4 such that p2´14 is not of the form 4ap8b ` 7q for

a, b P N and let O be the Z“

1p

‰

-order16

O “

x P Z“

1p

‰

` Z“

1p

‰

i` Z“

1p

‰

j ` Z“

1p

‰

k : x ” 1 mod 2(

in DQp . Let Γ be a finite index subgroup of Γ1O . Then as sÑ `8,

pp` 1q2 rΓ1O : Γs

2 p2 kΓs´1

ÿ

α PΓ¨α0, hpαqďs

∆α˚á HaarQp ,

where kΓ is the smallest positive integer such that„

1` ε x1 ´x3 ` ε x2

x3 ` ε x2 1´ ε x1

kΓ

P Γ. Further-

more, there exists κ ą 0 such that as sÑ `8

Cardtα P pΓ ¨ α0q X Zp : hpαq ď su “2 p2 kΓ

pp` 1q2 rΓ1O : Γs

s`Ops1´κq .

Proof. The group Oˆ of invertible elements of O is

Oˆ “

x P O : Npxq P pZ(

.

The centre of Oˆ is ZpOˆq “ t˘pn : n P Zu and the centre of O1 is ZpO1q “ t˘1u. Weidentify O1ZpO1q with its image in OˆZpOˆq. The quotient group OˆZpOˆq is a freegroup on s “ p`1

2 generators γ1, γ2, . . . , γs, which are the images modulo ZpOˆq of someelements of O of norm p, see for instance [Lub2, Coro. 2.1.11].17

Since Nppq “ p2, any reduced word of even length in S “ tγ˘1 , γ˘2 , . . . , γ

˘s u belongs to

O1ZpO1q. Two distinct elements in S differ by a reduced word of length 2, and γ1 does notbelong to O1ZpO1q. Hence t1, γ1u is a system of left coset representatives of O1ZpO1q inOˆZpOˆq, and the index of O1ZpO1q in OˆZpOˆq is

rOˆZpOˆq : O1ZpO1qs “ 2 . (17.10)

Let

g0 “

˜

1`ε x1p

´x3`ε x2p

x3`ε x2p

1´ε x1p

¸

.

By the definition of the isomorphism θ in Equation (17.8) (with?a “

?b “ ε) and of the

integers x1, x2, x3, the element g0 belongs to θpOq since x1, x2, x3 are even (and p is odd), anddet g0 “ 1. Hence g0 P θpO1q. Its fixed points for its action by homography on P1pQpq are,by an easy computation,

ε x1 ˘a

1´ p2

x3 ` ε x2.

16This order plays an important role in the construction of Ramanujan graphs by Lubotzky, Phillips andSarnak [LuPS1, LuPS2] (see also [Lub2, §7.4]), and in the explicit construction of free subgroups of SOp3qin order to construct Hausdorff-Banach-Tarsky paradoxical decompositions of the 2-sphere, see for instance[Lub2, page 11].

17The group OˆZpOˆq is denoted by Λp2q in [Lub2, page 11].

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In particular, α0 is one of these two fixed points. Note that tr g0 “2p , hence |vpptr g0q| “ 1,

and the image rg0s of g0 in PGL2pQpq is a primitive loxodromic element of Γ1O .

Let us defineγ0 “ rg0s

ε kΓ

where ε P t˘1u is chosen so that γ´0 “ α0 and where kΓ is defined in the statement of Theorem17.10. Since Γ has finite index in Γ1

O , some power of rg0s does belong to Γ, hence kΓ exists(and note that kΓ “ 1 if Γ “ Γ1

O). By the minimality of kΓ, the element γ0 is a primitiveloxodromic element of Γ. We will apply Theorem 17.1 to this γ0.

The algebra isomorphism θ induces a group isomorphism from OˆZpOˆq onto its imagein PGL2pQpq, that we denote by ΓˆO .

18 By [Lub2, Lem. 7.4.1], the group ΓˆO acts simplytransitively on the vertices of the Bruhat-Tits tree Xp.

In particular, Γ1O acts freely on Xp, and by Equation (17.10), we have

VolpΓ1OzzXpq “ rΓ

ˆO : Γ1

O s VolpΓˆOzzXpq“ rOˆZpOˆq : O1ZpO1qs Card

`

ΓˆOzV Xp˘

“ 2 . (17.11)

Again since Γ1O (hence Γ) acts freely on Xp, and since γ0 is primitive loxodromic in Γ, we

have

VolpΓγ´0zzAxγ0q “ Card

`

Γγ´0zVAxγ0

˘

“ λpγ0q

“ kΓ λprg0sq “ 2 kΓ |vpptr g0q| “ 2 kΓ . (17.12)

Using Equations (17.11) and (17.12), the result now follows from Theorem 17.1. l

18This group is denoted by Γp2q in [Lub2, page 95].

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Chapter 18

Counting and equidistribution ofcrossratios

We use the same notation as in Chapter 17: Kv is a non-Archimedean local field, withvaluation v, valuation ring Ov, choice of uniformiser πv, residual field kv of order qv, and Xvis the Bruhat-Tits tree of pPGL2,Kvq. Let Γ be a lattice in PGL2pKvq.

In this Chapter, we give counting and equidistribution results in Kv “ B8Xv ´ t8u oforbit points under Γ, using a complexity defined using crossratios, which is different from theone in Chapter 17. We refer to [PaP14b] for the development when Kv is R or C with itsstandard absolute value.

Recall that the crossratio of four pairwise distinct points a, b, c, d in P1pKvq “ Kv Y t8u

is

ra, b, c, ds “pc´ aq pd´ bq

pc´ bq pd´ aqP pKvq

ˆ ,

with the standard conventions when one of the points is8. Adopting Ahlfors’s terminology inthe complex case, the absolute crossratio of four pairwise distinct points a, b, c, d P P1pKvq “

Kv Y t8u is

|a, b, c, d|v “ |ra, b, c, ds|v “|c´ a|v |d´ b|v|c´ b|v |d´ a|v

,

with conventions analogous to the previous ones when one of the points is 8. As in theclassical case, the crossratio and the absolute crossratio are invariant under the diagonalprojective action of GL2pKvq on the set of quadruples of pairwise distinct points in P1pKvq.

18.1 Counting and equidistribution of crossratios of loxodromicfixed points

Let α, β P Kv be loxodromic fixed points of Γ. Recall that ασ, βσ is the other fixed point ofa loxodromic element of Γ fixing α, β, respectively. The relative height of β with respect to αis1

hαpβq “1

|α´ ασ|v |β ´ βσ|vmax

|β ´ α|v |βσ ´ ασ|v, |β ´ α

σ|v |βσ ´ α|v

(

.

1The factor |α´ ασ|v in the denominator, that did not appear in [PaP14b] in the analogous definition forthe case when Kv is R or C, is there in order to simplify the statements below.

273 19/12/2016

When β R tα, ασu, we have

hαpβq “ maxt|α, β, βσ, ασ|v, |α, βσ, β, ασ|vu “

1

mint|α, β, ασ, βσ|v, |α, βσ, ασ, β|vu. (18.1)

We will use the relative height as a complexity when β varies in a given orbit of Γ (and α isfixed).

The following properties of relative heights are easy to check using the definitions and theinvariance properties of the crossratio.

Lemma 18.1. Let α, β P Kv be loxodromic fixed points of Γ. Then(1) hαρpβτ q “ hαpβq for all ρ, τ P tid, σu.(2) If β P tα, ασu, then hαpβq “ 1.(3) hγ¨αpγ ¨ βq “ hαpβq for every γ P Γ.(4) hαpγ ¨ βq “ hαpβq for every γ P StabΓptα, α

σuq. l

The following result relates the relative height of two loxodromic fixed points with thedistance between the two translation axes.

Proposition 18.2. Let α, β P Kv be loxodromic fixed points of Γ such that β R tα, ασu. Then

hαpβq “ q dpsα,ασr , sβ,βσrq

v .

In particular, we have hαpβq ą 1 if and only if the geodesic lines sα, ασr and sβ, βσr in Xvare disjoint, and hαpβq “ 1 otherwise (using Lemma 18.1 (2) when β P tα, ασu).

Proof. Up to replacing α, β, ασ, βσ by their images under a big enough power γ of a loxo-dromic element in Γ with attracting fixed point in Ov, we may assume that these four pointsbelong to Ov. Note that γ exists since ΛΓ “ B8Xv, and it preserves the relative height byLemma 18.1 (3) as well as the distances between translation axes.

Let A “ sα, ασr and B “ sβ, βσr. Let u be the closest point to ˚v on A, so that

vpα´ ασq “ dpu, ˚vq .

We will consider five configurations.

α ασ β βσ

˚v

β βσα ασ

˚v

a “ u b

u1

b

u

a

Case 1. First assume that A and B are disjoint. Let ra, bs be the common perpendicularfrom A to B, with a P A, so that

dpA,Bq “ dpa, bq .

274 19/12/2016

First assume that u ‰ a. Up to exchanging α, ασ (which does not change dpA,Bq or hαpβqby Lemma 18.1 (1)), we may assume that a P ru, αr . Then (see the picture on the left above),

vpβ ´ βσq “ dpb, ˚vq, vpα´ βq “ vpα´ βσq “ dpa, ˚vq

andvpασ ´ βq “ vpασ ´ βσq “ dpu, ˚vq .

Therefore

|α, β, ασ, βσ|v “ |α, βσ, ασ, β|v “ q vpα´βq`vpα

σ´βσq´vpα´ασq´vpβ´βσqv

“ qdpa,˚vq´dpb,˚vqv “ q´dpa,bqv “ q´dpA,Bqv ,

which proves the result by Equation (18.1).

Assume on the contrary that u “ a. Let u1 P V Xv be such that ra, ˚vs X ra, bs “ ra, u1s.Note that u1 P r˚v, bs since β, βσ P Ov. Then (see the picture on the right above),

vpβ ´ βσq “ dpb, ˚vq, vpα´ βq “ vpα´ βσq “ vpασ ´ βq “ vpασ ´ βσq “ dpu1, ˚vq .

Therefore

|α, β, ασ, βσ|v “ |α, βσ, ασ, β|v “ q vpα´βq`vpα

σ´βσq´vpα´ασq´vpβ´βσqv

“ q2dpu1,˚vq´dpa,˚vq´dpb,˚vqv “ q´dpa, bqv “ q´dpA,Bqv ,

which proves the result by Equation (18.1).

βσ ασα β

˚v

α

˚v

β βσασα

˚v

β ασβσ

b

u

a a

u1

u “ bba

u

Case 2. Now assume that A and B are not disjoint, so that

dpA,Bq “ 0 .

Since β R tα, ασu, the intersection A X B is a compact segment ra, bs (possibly with a “ b)in Xv. Up to exchanging α and ασ, as well as β and βσ (which does not change dpA,Bq norhαpβq by Lemma 18.1 (1)), we may assume that α, a, b, ασ and β, a, b, βσ are in this order onA and B respectively, and that a P ru, αr .

Assume first that b P su, αr . Then (see the picture on the left above),

vpα´ βq “ dpa, ˚vq, vpα´ βσq “ vpβ ´ βσq “ dpb, ˚vq

andvpβ ´ ασq “ vpβσ ´ ασq “ dpu, ˚vq .

275 19/12/2016

Therefore

|α, βσ, ασ, β|v “ q vpα´βq`vpασ´βσq´vpα´ασq´vpβ´βσq

v “ qdpa,˚vq´dpb,˚vqv “ qdpa,bqv ě 1 ,

and|α, β, ασ, βσ|v “ q vpα´β

σq`vpασ´βq´vpα´ασq´vpβ´βσqv “ q0

v “ 1 “ q´dpA,Bqv ,

which proves the result by Equation (18.1).

Assume that b P su, ασr . Then (see the picture in the middle above),

vpα´ βq “ dpa, ˚vq, vpασ ´ βσq “ dpb, ˚vq

andvpα´ βσq “ vpβ ´ ασq “ vpβ ´ βσq “ dpu, ˚vq .

Therefore

|α, βσ, ασ, β|v “ q vpα´βq`vpασ´βσq´vpα´ασq´vpβ´βσq

v

“ qdpa,˚vq`dpb,˚vq´2dpu,˚vqv “ qdpa,bqv ě 1 ,

and|α, β, ασ, βσ|v “ q vpα´β

σq`vpασ´βq´vpα´ασq´vpβ´βσqv “ q0

v “ 1 “ q´dpA,Bqv ,

which proves the result by Equation (18.1).

Assume at last that b “ u. Let u1 P V Xv be such that rb, ˚vs X rb, βσr “ rb, u1s. Then (seethe picture on the right above),

vpα´ βq “ dpa, ˚vq, vpασ ´ βq “ dpu, ˚vq

andvpα´ βσq “ vpβ ´ βσq “ vpασ ´ βσq “ dpu1, ˚vq .

Therefore

|α, βσ, ασ, β|v “ q vpα´βq`vpασ´βσq´vpα´ασq´vpβ´βσq

v

“ qdpa,˚vq´dpu,˚vqv “ qdpa,bqv ě 1 ,

and|α, β, ασ, βσ|v “ q vpα´β

σq`vpασ´βq´vpα´ασq´vpβ´βσqv “ q0

v “ 1 “ q´dpA,Bqv ,

which proves the result by Equation (18.1). l

The next result says that the relative height is an appropriate complexity on a given orbitunder Γ of a loxodromic fixed point, and that the counting function we will study is welldefined. We denote by Γξ the stabiliser in Γ of a point ξ P B8Xv “ P1pKvq.

Lemma 18.3. Let α, β P Kv be loxodromic fixed points of Γ. Then for every s ą 1, the set

Es “ tβ1 P ΓαzΓ ¨ β : 1 ă hαpβ

1q ď su

is finite.

276 19/12/2016

Proof. The set Es is well defined by Lemma 18.1 (4). Recall that a loxodromic fixed point isone of the two points at infinity of a unique translation axis. By local finiteness, there are, upto the action of the stabiliser of a fixed translation axis A, only finitely many images underΓ of another translation axis B at distance at most ln s

ln qvfrom A. Since the stabiliser of A

contains the stabiliser of either of its points at infinity with index at most 2, the result thenfollows from Proposition 18.2. l

We now state our main counting and equidistribution result of orbits of loxodromic fixedpoints, when the complexity is the relative height with respect to a fixed loxodromic fixedpoint.

Theorem 18.4. Let Γ be a lattice in PGL2pKvq`. Let α0, β0 P Kv be loxodromic fixed points

of Γ. Then for the weak-star convergence of measures on Kv ´ tα0, ασ0u, as sÑ `8,

pqv ` 1q2 VolpΓzzXvq2 q2

v |α0 ´ ασ0 |v VolpΓβ0zz sβ0, βσ0 rqs´1

ÿ

βPΓ¨β0 : hα0 pβqďs

∆β˚á

dHaarKvpzq

|z ´ α0|v |z ´ ασ0 |v.

Furthermore, as sÑ `8,

Card Γα0ztβ P Γ ¨ β0 : hα0pβq ď su „2 qv VolpΓα0zz sα0, α

σ0 rq VolpΓβ0zz sβ0, β

σ0 rq

pqv ` 1q VolpΓzzXvqs .

If Γ is geometrically finite, for every β1 P s0, 1s, there exists κ ą 0 such that for everyψ P C β1

c pKv´tα0, ασ0uq, where Kv´tα0, α

σ0u is endowed with the distance-like map dsα0, ασ0 r

,2

there is an error term in the equidistribution claim of Theorem 18.4 when evaluated on ψ, ofthe form Ops´κψβ1q. This result applies for instance if ψ : Kv ´ tα0, α

σ0u Ñ R is locally

constant with compact support, see Remark 3.2.

Proof. The proof of the equidistribution claim is similar to the one of Theorem 17.1. Wenow apply Theorem 15.4 with D´ :“ sα0, α

σ0 r and D` :“ sβ0, β

σ0 r . Since Γ is contained in

PGL2pKvq`, the length spectrum LΓ of Γ is equal to 2Z. The families D˘ “ pγD˘qγPΓΓD˘

are locally finite, and σ´D` is finite and nonzero.Arguing as in the proof of Theorem 17.1,3 we have

limnÑ`8

pqv2 ´ 1qpqv ` 1q

2 qv3

VolpΓzzXvqσ´D`

qv´n

ÿ

γPΓΓD`0ădpD´, γD`qďn

∆γ¨β0 “ pB`q˚rσ`

D´ (18.2)

for the weak-star convergence of measures on B8Xv ´ B8D´. When Γ is geometrically finite,for every β1 P s0, 1s, there exists κ ą 0 such that for every β-Hölder-continuous functionψ P C β1

c pB8Xv ´ B8D´q, where B8Xv ´ B8D´ is endowed with the distance-like map dD´ ,there is an error term in the equidistribution claim of Theorem 18.4 when evaluated on ψ, ofthe form Ops´κψβ1q.

By Proposition 18.2, we have

hα0pγ ¨ β0q “ q dpD´, γD`q

v .

2See Equation (15.14).3See Equation (17.2).

277 19/12/2016

By Proposition 15.2 (5), we have

pB`q˚rσ`

D´ “|α0 ´ α

σ0 |v

|z ´ α0|v |z ´ ασ0 |vdHaarKvpzq

on the full measure subset Kv´tα0, ασ0u of B8Xv. Hence, using the change of variable s “ q nv ,

we have, with the appropriate error term when Γ is geometrically finite,

limsÑ`8

pqv2 ´ 1qpqv ` 1q

2 q3v

VolpΓzzXvq|α0 ´ ασ0 |v σ

´

D`s´1

ÿ

γPΓΓD`1ăhα0 pγ¨β0qďs

∆γ¨β0 “dHaarKvpzq

|z ´ α0|v |z ´ ασ0 |v.

We again denote by ια0 the index

ια0 “ rΓtα0, ασ0 u: Γα0s ,

and similarly for β0. Since the stabiliser Γβ0 of β0 in Γ has index ιβ0 in ΓD` and ΓΓβ0

identifies with Γ ¨ β0 by the map γ ÞÑ γ ¨ β0, we have, with the appropriate error term whenΓ is geometrically finite,

limsÑ`8

pqv2 ´ 1qpqv ` 1q

2 q3v

VolpΓzzXvq|α0 ´ ασ0 |v σ

´

D` ιβ0

s´1ÿ

βPΓ¨β0

1ăhα0 pβqďs

∆β “dHaarKvpzq

|z ´ α0|v |z ´ ασ0 |v.

As in the end of the proof of Theorem 17.1, we have

σ´D` “qv ´ 1

qv ιβ0

VolpΓβ0zz sβ0, βσ0 rq .

This proves the equidistribution claim, and its error term when Γ is geometrically finite.In order to obtain the counting claim, we note that since rσ`D´ is invariant under the

stabiliser in Γ of D´, hence under Γα0 , the measures on both sides of the equidistributionclaim in Theorem 18.4 are invariant under Γα0 , see Proposition 17.3 (5) for the invariance ofthe right hand side. By Proposition 15.2 (5) and (6), and by the definition of ια0 , we have

ż

Γα0zpKv´tα0, ασ0 uq

dHaarKvpzq

|z ´ α0|v |z ´ ασ0 |v“

ια0

|α0 ´ ασ0 |v

ż

ΓD´zB1`D´

drσ`D´

“pqv ` 1q ια0 VolpΓD´zzD´q

qv |α0 ´ ασ0 |v

“pqv ` 1q VolpΓα0zz sα0, α

σ0 rq

qv |α0 ´ ασ0 |v. (18.3)

The counting claim follows by evaluating the equidistribution claim on the characteristicfunction ψ of a compact open fundamental domain for the action of Γα0 on Kv ´ tα0, α

σ0u.

This characteristic function is locally constant, hence β1-Hölder-continuous for the distance-like function dD´ , as seen end of Section 3.2. l

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18.2 Counting and equidistribution of crossratios of quadraticirrationals

In this Section, we give two arithmetic applications of Theorem 18.4.

Let us first consider an application in positive characteristic. Let K be a function fieldover Fq, let v be a (normalised discrete) valuation of K, and let Rv be the affine functionring associated with v. We assume that the characteristic of K is different from 2. Given twoquadratic irrationals α, β P Kv over K, with Galois conjugates ασ, βσ respectively, such thatβ R tα, ασu, we define the relative height of β with respect to α by

hαpβq “1

mint|α, β, ασ, βσ|v, |α, βσ, ασ, β|vu. (18.4)

The following result says that the orbit of any quadratic irrational in Kv over K, by homo-graphies under a given finite index subgroup of the modular group PGL2pRvq, equidistributes,when its complexity is given by the relative height with respect to another fixed quadraticirrational α0. The limit measure is absolutely continuous with respect to the Haar measureon Kv and it is invariant under the stabiliser of α0 in PGL2pRvq by Proposition 17.3 (5).

Theorem 18.5. Let G be a finite index subgroup of GL2pRvq. Let α0, β0 P Kv be quadraticirrationals over K. Then, as sÑ `8,

pqv ` 1q2 ζKp´1q n0 rGL2pRvq : Gs

2 q2v pq ´ 1q |α0 ´ ασ0 |v |vptrh0q|

s´1ÿ

βPG¨β0 : hα0 pβqďs

∆β˚á

dHaarKvpzq

|z ´ α0|v |z ´ ασ0 |v,

and there exists κ ą 0 such that, as sÑ `8,

Card Γα0ztβ P G ¨ β0 : hα0pβq ď su “4 qv pq ´ 1q |vptr g0q| |vptrh0q| |ZpGq|

pqv ` 1q ζKp´1q m0 n0 rGL2pRvq : Gss`Ops1´κq .

Here g0, h0 P G fixes α0, β0 with vptr g0q, vptrh0q ‰ 0, and m0, n0 is the index of gZ0 , hZ0 in the

stabiliser of α0, β0 in G respectively.

Proof. This follows, as in the proof of Theorem 17.5, from Theorem 18.4 using Equations(16.6) and (17.7), as well as Equation (18.3) for the counting claim. l

Example 18.6. (1) Theorem 1.16 in the Introduction follows from this result, by takingK “ FqpY q and v “ v8, and by using Equation (14.5) in order to simplify the constant.

(2) If GI is the Hecke congruence subgroup associated with a nonzero ideal I of Rv (seeEquation (16.11)), using Lemma 16.5, we have, as sÑ `8,

pqv ` 1q2 ζKp´1q n0 NpIqś

p|Ip1`1

Nppqq

2 q2v pq ´ 1q |α0 ´ ασ0 |v |vptrh0q|

s´1ÿ

βPGI ¨β0 : hα0 pβqďs

∆β˚á

dHaarKvpzq

|z ´ α0|v |z ´ ασ0 |v.

The second arithmetic application of Theorem 18.4 is in Qp. We use the notation ofSection 17.3.

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Let p P N be a positive rational prime with p ” 1 mod 4 such that p2´14 is not of the

form 4ap8b ` 7q for a, b P N (for instance p “ 5). Let ε be a square root of ´1 in Qp. Letx1, x2, x3 P 2Z be such that p2 ´ 1 “ x1

2 ` x22 ` x3

2. We again consider

α0 “ε x1 `

a

1´ p2

x3 ` ε x2,

which is a quadratic irrational in Qp over Qpεq. We denote by ασ the Galois conjugate of aquadratic irrational α in Qp over Qpεq, and by

hαpβq “1

mint|α, β, ασ, βσ|p, |α, βσ, ασ, β|pu(18.5)

the relative height of a quadratic irrational β in Qp over Qpεq with respect to α, such thatβ R tα, ασu. We again consider Hamilton’s quaternion algebra D “ p

´1,´1Q q over Q and its

Z“

1p

‰

-order

O “

x P Z“

1p

‰

` Z“

1p

‰

i` Z“

1p

‰

j ` Z“

1p

‰

k : x ” 1 mod 2(

.

The following result says that the orbit of α0 in Qp by homographies under a given finiteindex subgroup of the arithmetic group Γ1

O equidistributes, when its complexity is given bythe relative height with respect to α0, to a measure absolutely continuous with respect to theHaar measure on Qp.

Theorem 18.7. With the above notation, let Γ be a finite index subgroup of Γ1O . Then, as

sÑ `8,

pp` 1q2 rΓ1O : Γs

2 p2 kΓ |α0 ´ ασ0 |ps´1

ÿ

αPΓ¨α0 : hα0 pαqďs

∆α˚á

dHaarQppzq

|z ´ α0|p |z ´ ασ0 |p,

where kΓ is the smallest positive integer such that„

1` ε x1 ´x3 ` ε x2

x3 ` ε x2 1´ ε x1

kΓ

P Γ. Further-

more, there exists κ ą 0 such that, as sÑ `8,

Card Γα0ztα P Γ ¨ α0 : hα0pαq ď su “4 p pkΓq

2

pp` 1q rΓ1O : Γs

s`Ops1´κq .

Proof. This follows, as in the proof of Theorem 17.10, from Theorem 18.4 using Equations(17.11) and (17.12), as well as Equation (18.3) for the counting claim. l

280 19/12/2016

Chapter 19

Counting and equidistribution ofintegral representations by quadraticnorm forms

In the final Chapter of this text, we give another equidistribution and counting result ofrational elements in non-Archimedean local fields of positive characteristic, again using ourequidistribution and counting results of common perpendiculars in trees summarized in Sec-tion 15.4. In this Chapter, we use a complexity defined using the norm forms associated withfixed quadratic irrationals. In particular, the complexity in this Chapter is different from thatused in the Mertens type of results in Section 16.1. We refer for instance to [PaP14b, §5.3]for motivations and results in the Archimedean case, and also to [GoP] for higher dimensionalnorm forms.

Let K be a function field over Fq, let v be a (normalised discrete) valuation of K, and letRv be the affine function ring associated with v. Let α P Kv be a quadratic irrational overK. The norm form nα associated with α is the quadratic form K ˆK Ñ K defined by

px, yq ÞÑ npx´ yαq “ px´ yαqpx´ yασq “ x2 ´ xy trpαq ` y2 npαq .

See Proposition 17.3 for elementary transformation properties under elements of GL2pRvq ofthis norm form.

A pair px, yq P Rv ˆRv is an integral representation of an element z P K by the quadraticnorm form nα if nαpx, yq “ z. The following result describes the projective equidistributionas s Ñ `8 of the integral representations by nα of elements with absolute value at most s.For every px0, y0q P Rv ˆ Rv, let Hpx0,y0q be the stabiliser of px0, y0q for the linear action ofany subgroup H of GL2pRvq on Rv ˆ Rv. We use the notation Nxx0, y0y for the norm ofthe ideal xx0, y0y generated by x0, y0 (see Section 14.2) and the notation mv, x0, y0 introducedabove Theorem 16.1.

Theorem 19.1. Let G be a finite index subgroup of GL2pRvq, let α P Kv be a quadraticirrational over K, and let px0, y0q P Rv ˆRv ´ tp0, 0qu. Let

c1 “pqv ´ 1q pqv ` 1q2 ζKp´1q mv, x0, y0 pNxx0, y0yq

2 rGL2pRvq : Gs

q3v pq ´ 1q qg´1 rGL2pRvqpx0, y0q : Gpx0, y0qs

.

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Then for the weak-star convergence of measures on Kv ´ tα, ασu, we have

limsÑ`8

c1 s´1ÿ

px, yqPGpx0, y0q, | npx´yαq|vďs

∆xy“

dHaarKvpzq

|z ´ α|v |z ´ ασ|v.

For every β P s0, 1s, there exists κ ą 0 such that for every ψ P C βc pKv ´ tα, α

σuq,where Kv ´ tα, α

σu is endowed with the distance-like map dsα, ασr,1 as for instance if ψ :Kv ´ tα, α

σu Ñ R is locally constant with compact support (see Remark 3.2), there is anerror term in the equidistribution claim of Theorem 19.1 when evaluated on ψ, of the formOps´κψβq.

Examples 19.2. (1) Let px0, y0q “ p1, 0q, K “ FqpY q and v “ v8. Theorem 1.17 in theIntroduction follows from Theorem 19.1, using Equations (14.5) and (16.1) to simplify theconstant c1.(2) Let px0, y0q “ p1, 0q and let G “ GI be the Hecke congruence subgroup of GL2pRvq definedin Equation (16.11). The index in rGL2pRvq : GIs is given by Lemma 16.5 and GI satisfiespGIqp1,0q “ GL2pRvqp1,0q. For every nonzero ideal I of Rv, for the weak-star convergence ofmeasures on Kv ´ tα, α

σu, we have

limsÑ`8

cI s´1

ÿ

px, yqPRvˆI, xx, yy“Rv , | npx´yαq|vďs

∆xy“

dHaarKvpzq

|z ´ α|v |z ´ ασ|v,

where

cI “pqv ´ 1q pqv ` 1q2 ζKp´1q NpIq

ś

p|Ip1`1

Nppqq

q3v q

g´1.

(3) This example is interesting when the ideal class number is larger than 1. Given anyfractional ideal m of Rv, taking px0, y0q P Rv ˆRv such that the fractional ideals xx0, y0y andm have the same ideal class and G “ GL2pRvq, using the change of variables s ÞÑ sNpmq2 inthe statement of Theorem 19.1, for the weak-star convergence of measures on Kv ´ tα, α

σu,with the same error term as for Theorem 19.1, we have

limsÑ`8

cm s´1

ÿ

px, yqPmˆm, xx, yy“m, Npmq´2| npx´yαq|vďs

∆xy“

dHaarKvpzq

|z ´ α|v |z ´ ασ|v,

where

cm “pqv ´ 1q pqv ` 1q2 ζKp´1q mv, x0, y0

q3v pq ´ 1q qg´1

.

Before proving Theorem 19.1, let us give a counting result which follows from it. Anysubgroup of G acts on the left on any orbit of G. Furthermore, the stabiliser Gα of α inG preserves the map px, yq ÞÑ | npx ´ yαq|v, by Proposition 17.3 (3). We may then define acounting function Ψ1psq “ Ψ1G,α, x0,y0

psq of elements in RvˆRv in a linear orbit under a finiteindex subgroup G of GL2pRvq on which the absolute value of the norm form associated withα is at most s, as

Ψ1psq “ Card Gα z

px, yq P Gpx0, y0q, | npx´ yαq|v ď su .

1See Equation (15.14).

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Corollary 19.3. Let G be a finite index subgroup of GL2pRvq, let α P Kv be a quadraticirrational over K, and let px0, y0q P Rv ˆRv ´ tp0, 0qu. Let g0 P Gα with vptr g0q ‰ 0 and letm0 be the index of gZ0 in Gα. Let

c2 “2 q2

v pq ´ 1q qg´1 |ZpGq| |vptr g0q| rGL2pRvqpx0, y0q : Gpx0, y0qs

pqv2 ´ 1q ζKp´1q |α´ ασ|v m0 mv, x0, y0 pNxx0, y0yq2 rGL2pRvq : Gs

.

Then there exists κ ą 0 such that, as sÑ `8,

Ψ1psq “ c2 s`Ops1´κq .

Proof. Using Equation (18.3) (with Γ the image of G in PGL2pRvq) and Equation (17.7),we have

ż

GαzpKv´tα, ασuq

dHaarKvpzq

|z ´ α|v |z ´ ασ|v“

2 pqv ` 1q |ZpGq| |vptr g0q|

qv |α´ ασ|v m0.

The corollary then follows by applying the equidistribution claim in Theorem 19.1 to thecharacteristic function of a compact-open fundamental domain of Kv ´ tα, α

σu modulo theaction by homographies of Gα. l

Example 19.4. Let px0, y0q “ p1, 0q, K “ FqpY q, v “ v8 and G “ GL2pFqrY sq. UsingEquations (14.5) and (16.1), Proposition 17.3 (1), and the fact that |ZpGq| “ q´1 to simplifythe constant c2 of Corollary 19.3, and recalling the expression of the absolute value at 8 interms of the degree from Section 14.2, we get the following counting result: For every integralquadratic irrational α P FqppY qq over FqpY q, there exists κ ą 0 such that, as tÑ `8,

Card GL2pFqrY sqα

I

"

px, yq PFqrY s ˆ FqrY s :xx, yy “ FqrY s,degpx2 ´ xy trpαq ` y2 npαqq ď t

*

“2

m0pq ´ 1q4 degptr g0q q

1´ 12

degptrpαq2´4 npαqq qt `Opqt´κq ,

where g0 P GL2pFqrY sq fixes α with degptr g0q ‰ 0 and m0 is the index of gZ0 in the stabiliserGL2pFqrY sqα of α in GL2pFqrY sq.

Proof of Theorem 19.1. The proof is similar to that of Theorem 16.1. Let r “ x0y0P

K Y t8u. If y0 “ 0, let gr “ id P GL2pKq, and if y0 ‰ 0, let

gr “

ˆ

r 11 0

˙

P GL2pKq .

We apply Theorem 15.4 with Γ :“ G the image of G in PGL2pRvq, D´ :“ sα, ασr the (imageof any) geodesic line in Xv with points at infinity α and ασ, and D` :“ γrH8, where γr isthe image of gr in PGL2pRvq.

We have LΓv “ 2Z and the family D` “ pγD`qγPΓΓD`is locally finite, as seen in the

beginning of the proof of Theorem 16.1. The family D´ “ pγD´qγPΓΓD´is locally finite as

seen in the beginning of the proof of Theorem 17.1.By Proposition 15.2 (5), we have (on the full measure subset Kv ´ tα, α

σu of B8Xv)

pB`q˚rσ`

D´ “|α´ ασ|v

|z ´ α|v |z ´ ασ|vdHaarKvpzq .

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For every γ P ΓΓr such that D´ and γD` are disjoint, let ργ be the geodesic ray startingfrom α´e, γp0q and ending at the point at infinity γ ¨ r of γD`.

Hence, as in order to obtain Equation (16.2), we have, with an error term for everyβ P s0, 1s of the form Ops´κψβq for some κ ą 0 when evaluated on ψ P C β

c pB8Xv ´ B8D´q,

limnÑ`8

pqv2 ´ 1qpqv ` 1q

2 q3v

VolpΓzzXvqσ´D`

qv´n

ÿ

γPΓΓr0ădpD´, γD`qďn

∆γ¨r

“|α´ ασ|v

|z ´ α|v |z ´ ασ|vdHaarKvpzq . (19.1)

We use the following Lemma to switch from counting over elements γ P ΓΓr for which0 ă dpD´, γD`q ď t to counting over integral representations with bounded value of the normform. See [PaP11a, page 1054] for the analogous result for the real hyperbolic 3-space andindefinite binary Hermitian forms.

Lemma 19.5. Let g P GL2pRvq and let γ be the image of g in PGL2pKq. Let z0 “ y0 ify0 ‰ 0 and z0 “ x0 otherwise. Let px, yq “ gpx0, y0q. If dpD´, γD`q ą 0, then

dpD´, γD`q “1

ln qvln´

| npx´ yαq|vhpαq

|z0|2v

¯

.

Proof. We start by showing that

g grp1, 0q “` x

z0,y

z0

˘

.

Indeed, if y0 ‰ 0, we have

g grp1, 0q “ gpr, 1q “1

y0gpx0, y0q

and otherwiseg grp1, 0q “ gp1, 0q “

1

x0gpx0, 0q “

1

x0gpx0, y0q .

In particular,

pg grq´1 “

ˆ

˚ ˚

´yz0

xz0

˙

.

Note that g gr P GL2pKq and |detpg grq|v “ |det g|v | det gr|v “ 1 since g P GL2pRvq. ByProposition 17.3 (2), we hence have

hppg grq´1 ¨ αq “

ˇ

ˇ

ˇn` x

z0´y

z0α˘

ˇ

ˇ

ˇ

vhpαq “ | npx´ y αq|v

hpαq

|z0|2v

. (19.2)

We use the signed distance dpL,Hq “ minxPL βξpx, xHq between a geodesic line L anda horoball H centred at ξ ‰ L˘, where xH is any point of the boundary of H. Now, byEquations (15.2) and (2.8), we have

dpD´, γD`q “ dpsα, ασr , γγrH8q “ d`

spg grq´1 ¨ α, pg grq

´1 ¨ ασr ,H8

˘

“ v`

pg grq´1 ¨ α´ pg grq

´1 ¨ ασ˘

“´ ln

ˇ

ˇpg grq´1 ¨ α´ pg grq

´1 ¨ ασˇ

ˇ

v

ln qv“

lnhppg grq´1 ¨ αq

ln qv. (19.3)

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Combining Equations (19.2) and (19.3) gives the result. l

By discreteness, there are only finitely many double classes rgs P GαzGGpx0,y0q such thatD´ “ sα, ασr and gD` “ g grH8 are not disjoint. Let ZpGq be the centre of G, which is finite.Since ZpGq acts trivially on P1pKvq, the map GGpx0,y0q Ñ ΓΓr induced by the canonicalmap GL2pRvq Ñ PGL2pRvq is |ZpGq|-to-1. Using the change of variable

s “|z0|v

2

hpαqqvn ,

by using Lemma 19.5, Equation (19.1) gives

limsÑ`8

pqv2 ´ 1q pqv ` 1q |z0|v

2

2 qv3 |ZpGq|

VolpΓzzXvqσ´D`

s´1ÿ

px, yqPGpx0, y0q, | npx´yαq|vďs

∆xy

“dHaarKvpzq

|z ´ α|v |z ´ ασ|v,

with the appropriate error term. Replacing VolpΓzzXvq and σ´D` by their values respectivelygiven by Equation (16.6) and Lemma 16.3, the claim of Theorem 19.1 follows. l

285 19/12/2016

286 19/12/2016

Appendix A

A weak Gibbs measure is the uniqueequilibrium, by J. Buzzi

AbstractFor a transitive topological Markov shift endowed with a Hölder-con-tinuous potential, we prove that a weak Gibbs measure is the uniqueequilibrium measure.

A.1 Introduction

Let σ : Σ Ñ Σ be a topological Markov shift (possibly one- or two-sided), see for instanceSection 5.1. More precisely, we consider the one-sided and two-sided vertex-shifts defined bya countable oriented graph G with set of vertices VG and set of arrows AG Ă VG ˆ VG. Weassume that Σ is transitive, that is, that G is connected (as an oriented graph).

We denote by PpΣq the set of σ-invariant probability measures on Σ and by PergpΣq thesubset of ergodic ones. Recall that, for all n P N, the n-cylinders are the following subsets ofΣ, where x varies in Σ:

Cnpxq “ rx0 . . . xn´1s “ ty P Σ : @ k P t0, . . . , n´ 1u, yk “ xku ,

so that the 1-cylinders are rvs “ ty P Σ : y0 “ vu for all v P VG. The points of Σ admittingn P N as period under the shift σ form the set

FixnpΣq “ tx P Σ : σnx “ xu .

We fix a potential on Σ, that is, a continuous function φ : Σ Ñ R. We do not assume that φis bounded. We define φ´ “ maxt´φ, 0u and, for all n P N´ t0u,

varnpφq “ supx,yPΣ, @ kPt0,...,n´1u, xk“yk

|φpyq ´ φpxq|

if pΣ, σq is one-sided and otherwise

varnpφq “ supx,yPΣ, @ kPt´n`1,...,n´1u, xk“yk

|φpyq ´ φpxq|

We say that φ has summable variations ifř

ně1 varnpφq ă 8. Let Snφ “ 1n

řn´1i“0 φ ˝ σ

i forall n P N.

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Definition A.1. A weak Gibbs measure for the potential φ is a σ-invariant Borel probabilitymeasure m on Σ such that there exists a number cpmq P R such that for every v P VG, thereexists C ě 1 with

@ n ě 1, @x P FixnpΣq X rvs, C´1 ďmpCnpxqq

exp pSnφpxq ´ cpmqnqď C . (A.1)

Note that cpmq is then unique, called the Gibbs constant ofm. Let us stress that we do notassume the so-called Big Image Property [Sar1] and hence using the above weakened Gibbsproperty (that is, allowing C to depend on v) is necessary.

Note that if Σ is locally compact, that is, if every vertex of G has finite degree (the numberof arrows arriving or leaving from the given vertex), then the above condition is equivalent tothe fact that for any nonempty compact subset K in Σ, there exists C ě 1 with

@ n ě 1, @ x P FixnpΣq XK, C´1 ďmpCnpxqq

exp pSnφpxq ´ cpmqnqď C .

The pressure P pφ, νq of an element ν P PpΣq such thatş

φ´ dν ă `8 is

P pφ, νq “ hνpσq `

ż

φdν.

An equilibrium measure µeq for pΣ, φq is an element µeq P PpΣq such thatş

φ´ dµeq ă `8

andP pφ, µeqq “ suptP pφ, νq : ν P PpΣq and

ż

φ´ dν ă `8u .

The Gurevič pressure is

PGpφq “ limnÑ8

1

nln

ÿ

xPFixnpΣqXrvs

eSnφpxq

for any vertex v P VG. Note that the Gurevič pressure does not depend on v. Let us recall afew results on the above notions.

Theorem A.2 (Iommi-Jordan [IJ, Theorem 2.2]). The following variational principle holds:

PGpφq “ suptP pφ, νq : ν P PpΣq andż

φ´ dν ă `8u . l

Theorem A.3 (Buzzi-Sarig [BuS, Theorem 1.1]). If PGpφq ă 8, then there exists at mostone equilibrium measure.

If there exists an equilibrium measure µ, then dµ “ h dν where h : Σ Ñ R is a continuous,positive function and ν is a positive measure on Σ such that

‚ Lφ h “ ePGpφqh, and L˚φ ν “ ePGpφqν where Lφ is the transfer operator defined byLφ u pxq “

ř

yPσ´1x eφpyq upyq.

‚ ν is finite on each cylinder. l

We note that [BuS] assumed supφ ă 8, but this was only used to justify the variationalprinciple and so this condition can be removed by using Theorem A.2.

We now state the main result of this appendix.

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Theorem A.4. Let pΣ, σq be a one-sided transitive topological Markov shift and let φ : Σ Ñ Rbe a potential with summable variations. Let m be an invariant probability measure of Σ suchthat

ş

φ´dm ă `8.Then m is a weak Gibbs measure if and only if it is an equilibrium measure. In this

case, the Gibbs constant cpmq is equal to the Gurevič pressure and the equilibrium measure isunique.

By a classical argument that follows, this result extends to two-sided topological Markovshifts (up to a slight strengthening of the regularity assumption).

Corollary A.5. Let pΣ, σq be a two-sided transitive topological Markov shift and let φ : Σ Ñ Rbe a potential with

ř

ně1 n varnpφq ă 8. Let m be an invariant probability measure of Σ suchthat

ş

φ´dm ă `8.Then m is a weak Gibbs measure if and only if it is an equilibrium measure. In this

case, the Gibbs constant cpmq is equal to the Gurevič pressure and the equilibrium measure isunique.

Remark. The case of the full shift NZ has been treated in [PeSZ, Sec. 3]. More generally,assuming the Big Image Property, the above result follows from [Sar1] and [BuS] along thelines of [PeSZ].

Proof of Corollary A.5. Let pΣ, σq, φ, and m be as in the statement of this Corollary. Letπ : Σ Ñ Σ` with pxnqnPZ ÞÑ pxnqnPN be the obvious factor map onto the one-sided topologicalMarkov shift pΣ`, σ`q defined by the same graph G as for pΣ, σq, called the natural extension.

First, we replace φ by a potential φ depending only on future coordinates. The proof of[Bowe2, Lemma 1.6] applies to our non-compact setting without changes. To be more precise,for each vertex a P VG, choose za P Σ with za0 “ a, and define r : Σ Ñ Σ by rpxq “ y withyn “ xn for n ě 0 and yn “ zx0

n for n ď 0 and let

upxq “ÿ

kě0

pφ ˝ σk ´ φ ˝ σk ˝ rqpxq .

This defines a bounded real function on Σ since |φ ˝ σk ´ φ ˝ σk ˝ r| ď vark`1pφq and φ hassummable variations. Moreover, u itself has summable variations since, given x, y P Σ withxk “ yk for |k| ă n, we have

|upxq ´ upyq| ďÿ

0ďkďtn2u

´

|φpσkxq ´ φpσkyq| ` |φpσkprxqq ´ φpσkpryqq|¯

`ÿ

kątn2u

´

|φ ˝ σkpxq ´ φ ˝ σkprxq| ` |φ ˝ σkpyq ´ φ ˝ σkpryq|¯

ď 4ÿ

kěrn2s

vark`1pφq ,

so thatÿ

ně1

varnpuq ď 4ÿ

kě1

p2k ´ 2q varkpφq ă 8 .

Now define φ : Σ Ñ R byφ “ φ` u ˝ σ ´ u .

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The function φ is continuous with summable variations. Following [Bowe2], let us prove thatφ “ φ ˝ r. We have

φ “ φ`ÿ

kě0

pφ ˝ σk`1 ´ φ ˝ σk ˝ r ˝ σq ´ÿ

kě0

pφ ˝ σk ´ φ ˝ σk ˝ rq

“ φ´ φ´ÿ

kě0

´

φ ˝ σk ˝ r ˝ σ ´ φ ˝ σk ˝ r¯

“ÿ

kě0

´

φ ˝ σk ˝ r ´ φ ˝ σk ˝ r ˝ σ¯

.

Now, r2 “ r and r ˝ σ ˝ r “ r ˝ σ. Hence φ ˝ r “ φ as claimed. Thus, φ induces on theone-sided shift a function rφ : Σ` Ñ R defined by

φ : pxnqnPN ÞÑ φp. . . zx0´2z

x0´1x0x1 . . . q ,

satisfying φ “ rφ ˝ π.To conclude, observe that φ´ φ is bounded and that cylinders defined by the same finite

words have the same measure for an invariant probability measure m on the two-sided shiftpΣ, σq and for its image π˚m on the one-sided shift pΣ`, σ`q. Therefore m is a weak Gibbsmeasure for φ if and only if π˚m is a weak Gibbs measure for rφ.

By construction π˚mprφq “ mpφq “ mpφq since m is invariant. As it is well-known, thenatural extension π preserves the entropy. Thus, the measure m is an equilibrium measurewith respect to φ if and only if π˚m is an equilibrium measure with respect to rφ.

The reduction to one-sided topological Markov shifts is thus complete. l

A.2 Proof of the main result Theorem A.4

The uniqueness of the equilibrium state is given by Theorem A.3. We need to prove thatweak Gibbs measures and equilibrium measures coincide under the integrability assumptionon φ´ and that the number cpmq is equal to the pressure.

Step 1. If m is an equilibrium measure, then it is a weak Gibbs measure.

This is a routine consequence of Theorem A.3. Our definition of an equilibrium measurem enforces

ş

φ´ dm ă `8 (hences excludes the concomitance of hmpσq “ `8 andş

φdm “

´8).Recall from Theorem A.3 that dm “ h dν. For v P VG and x P FixnpΣq X rvs, we have

mpCnpxqq “

ż

h1Cnpxq dν “ e´nPGpφqż

Lnφphq1Cnpxq dν .

By definition, Lnφphq1Cnpxqpzq “ expSnφpx0 . . . xn´1zq hpx0 . . . xn´1zq for all z P σnpCnpxqq “rvs (and Lnφphq1Cnpxqpzq “ 0 otherwise). Hence

mpCnpxqq “ e´nPGpφq exp´

Snφpxq ˘nÿ

k“1

varkpφq¯

ż

rvsh dν .

As 0 ăş

rvs h dν ă `8 andř`8k“1 varkpφq ă `8, the measure m is a weak Gibbs measure for

φ with Gibbs constant cpmq “ PGpφq.

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We now turn to the converse implication. Let m be a weak Gibbs measure for φ such thatş

φ´ dm ă `8.

The weak Gibbs condition only controls the cylinders that start and end with the samesymbol. Passing to an induced system (that is, considering a first return map on a 1-cylinder)will remove this restriction. More precisely, let a P VG be a vertex of G and let µ be aninvariant probability measure on pΣ, σq with µprasq ą 0. The induced system on the 1-cylinderras “ tx P Σ : x0 “ au is the map σ : ras Ñ ras defined as follows:

‚ let τpxq “ inftn ě 1 : σnx P rasu be the first-return time in ras, that we also denoteby τraspxq when we want to emphasize ras;

‚ let σpxq “ στpxqpxq if τpxq ă 8;

‚ let µpBq “ µpB X rasqµprasq for every Borel subset B of Σ be the restriction of µ toras normalized to be a probability measure.

We also define τ0pxq “ 0 and by induction τn`1pxq “ τpxq ` τnpσxq for every n P N. Notethat σ can only be iterated on the subset

tx P ras : @ n ě 1, τnpxq ă 8u .

By Poincaré’s recurrence theorem, this is a full measure subset of ras, hence the distinctionwill be irrelevant for our purposes.

The induced partition is

β “ traξ1 . . . ξn´1as ‰ H : n ě 1, ξi ‰ au .

We note that σ : ras Ñ ras is topologically Bernoulli with respect to the partition β (that is,σ : bÑ ras is a homeomorphism for each b P β). For every integer N ě 1, we define the N -thiterated partition βN of β by

βN “ tb0 X σ´1b1 X ¨ ¨ ¨ X σ

´N`1bN´1 ‰ H : b0, . . . , bN´1 P βu

and we write βN pxq for the element of the partition βN that contains x.

Step 2. The topological Markov shift may be assumed to be topologically mixing.

This follows from the spectral decomposition for topological Markov shifts, see for instance[BuS, Lem. 2.2].

Step 3. The Gibbs property implies full support and ergodicity.

Let A be an invariant (σ´1pAq “ A) measurable subset of Σ with mpAq ą 0 and let usprove that mpAq “ 1.

Observe that the Gibbs property, together with the transitivity of Σ, implies that anycylinder has positive measure for m, hence that m has full support. Let a P VG be such thatmpAX rasq ą 0.

As mprasq ą 0, we may consider the induced system on ras. Let N ě 1. When f isa homeomorphism between topological spaces, let f˚ denote the pushforwards of measures

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by f´1. First note that, for almost every x P ras and every N P N ´ t0u, since σN is anhomeomorphism from βN pxq onto ras, we have

mpAX rasq

mprasq“mpσN pAX βN pxqqq

mpσN pβN pxqqq“

ş

βN pxqXAdpσN q˚m

dm dmş

βN pxqdpσN q˚m

dm dm.

Now, observe1 that for m-almost every y P βN pxq:

dpσN q˚m

dmpyq “ lim

nÑ8

mpστN pyqry0 . . . ynsq

mpry0 . . . ynsq“ C˘2e

´SτN pyq

φpyq`τN pyq cpmq.

Hence,mpAX rasq

mprasq“ C˘4 mpAX β

N pxqq

mpβN pxqq.

By Doob’s increasing martingale convergence theorem (see for instance [Pet]), for m-almostevery x P ras ´ A, the ratio on the right hand side converges to 0 as N Ñ 8. Thus ras iscontained in A modulo m. Therefore A “

Ť

aPW ras modulo m for some subset W of VG.Since Σ is topologically mixing, for any vertex b, the intersection ras X σ´irbs X σ´jras

is not empty for some integers 0 ă i ă j. Pick some point x in that set. By invariance,mprbsq ě mpσipCjpxqqq ě mpCjpxqq. But this last number is positive by the weak Gibbsproperty. Thus rbs is contained in A modulo m. Hence mpAq “ 1, proving the ergodicity ofm.

Step 4. The Gurevič pressure PGpφq is equal to cpmq, hence is finite. Furthermore hmpσq ă 8and φ P L1pmq.

Fix v P VG and let K “ tx P Σ : x0 “ vu. Note that mpKq ą 0. The ergodicity of m givesa Cesaro convergence: as nÑ8, we have

1

n

n´1ÿ

k“0

mpK X σ´kKq ÝÑ mpKq2 ą 0 .

The Gibbs property implies that, for all n ě 1,

mpK X σ´nKq “ C˘1ÿ

xPFixnpΣqXK

eSnφpxq´cpmqn

“ C˘1´

ÿ

xPFixnpΣqXK

eSnφpxq¯

e´cpmqn . (A.2)

If we write Zn for the term between the parenthesis, we have by the definition of the Gurevičpressure:

PGpφq “ lim supnÑ8

1

nlnZn .

As the value of the left hand side of Equation (A.2) is less than one, we see that cpmq ě PGpφq.If this was a strict inequality, then the left hand side of Equation (A.2) would converge tozero, contradicting its Cesaro convergence to mpKq2 ą 0. Therefore PGpφq “ cpmq.

Since cpmq is finite, so is PGpφq. Hence Theorem A.2 implies that, for any ν P PpΣq withş

φ´ dν ă `8, we have hνpσq ă 8 and φ is ν-integrable. In particular, this holds for ν “ m,which finishes the proof of Step 4.

1using, for all u, v, c ą 0 and n P N, the notation u “ c˘nv if 1cnv ď u ď cnv

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Step 5. If the mean entropy Hµpβq “ ´ř

bPβ µpbq lnµpbq is finite, then

hµpσq “ ´

ż

lndµ

dσ˚µdµ ,

where σ˚µ is the measure on Σ defined by B ÞÑř

bPβ µpσpB X bqq, with respect to which µ isabsolutely continuous: µ Î σ˚µ.

This is a classical formula which follows from the computation of the entropy in terms ofthe information function

hµpσq “ ´

ż

ÿ

bPβ

1bpxq limnÑ8

lnEµp1b | σ´1β _ ¨ ¨ ¨ _ σ´nβqpxq dµpxq

and from the identity, for x P b,

Eµp1b | σ´1β _ ¨ ¨ ¨ _ σ´nβqpxq “µpβn`1pxqq

µpβnpσxqq“

µpβn`1pxqq

σ˚µpβn`1pxqq.

The absolute continuity follows from a direct computation and ensures that the integral aboveis well-defined.

Step 6. For all a P VG, N ě 1 and µ P PergpΣq withş

φ´ dµ ă `8, we have

hµpσq “ ´µprasq

ż

ras

1

Nln

dµ

dppσN q˚ µqdµ (A.3)

We use arguments from the proof of [BuS, Theorem 1.1]: the key is to see that the partitionβ of ras has finite mean entropy for the induced measure µ using a Bernoulli approximation.

Let us consider the Bernoulli measure µB for pras, σq defined by

µB`

n´1č

i“0

σ´iBi˘

“

n´1ź

i“0

µpBiq

for all Bi P β. We construct from it an invariant and ergodic measure µB on pΣ, σq: For everyBorel subset A, let

µBpAq “ µprasq

ż

ras

τras´1ÿ

i“0

1A ˝ σi dµB .

Define φ “řτras´1

i“0 φ ˝σi. Using the assumption that φ has the summable variations, we haveż

φdµB ě µprasq

ż

rasφ dµ´ C “

ż

φdµ´ C ą ´8 .

Therefore, the last two lines of the proof of Step 4 apply to ν “ µB and hµB pσq ă `8. SinceµB is ergodic, Abramov’s formula yields

hµB pσq “ µprasqhµB pσq .

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Since µB is Bernoulli, the right hand side of this equality is equal to

µprasqHµB pβq “ µprasqHµpβq

which is hence proven to be finite. Thus, Step 5 applies:

hµpσq “ ´

ż

rasln

dµ

dpσ˚µqdµ .

This formula extends to hµpσN q for all integers N ě 1. Using Abramov’s formula this timefor µ and µ (since µ is ergodic), we have

hµpσq “µprasq

Nhµpσ

N q “ ´µprasq

ż

ras

1

Nln

dµ

dppσN q˚µqdµ ,

as claimed.

Step 7. The entropy of m is equal to cpmq ´ş

φdm.

In order to prove this, we apply Step 6 with µ “ m (which is possible, since m has beenproven to be ergodic in Step 3). As in the proof of Step 3, the Radon-Nikodym derivative isalmost everywhere

dµ

dppσN q˚µqpxq “ lim

nÑ8

µpβnpxqq

µpσN pβnpxqqq“ C˘2 exp

`

´ cpmqτN pxq ` SτN pxqφpxq˘

.

Therefore, using Step 6 and the fact that µ|ras “ µprasqµ, we have

hµpσq “ limNÑ8

1

N

´

ż

ras

`

cpmq τN pxq ´ SτN pxqφpxq˘

dµ˘ 2 lnC¯

. (A.4)

Note that τN pxq can be seen as a Birkhoff sum for the induced system on ras and the functionτ and that, by Kac’s theorem (see for instance [Pet, Sect. 2.4]),

ż

rasτ d µ “ µprasq´1 .

Therefore, Birkhoff’s ergodic theorem yields, with convergence in L1pµq,

limNÑ8

cpmq τN pxq

N“

cpmq

µprasq.

To analyze the second term in Equation (A.4), let φpxq “řτpxq´1k“0 φpσkxq and observe that,

by a variation of the proof of Kac’s theorem, φ P L1pµq with µpφq “ µprasq´1µpφq. Indeed,passing to the natural extension, one can assume the system to be invertible and use thepartition modulo µ given by

ď

ně1, 0ďkăn

σkptx P ras : τpxq “ nuq .

Since SτN pxqφpxq coincides with the Birkhoff sum SN φpxq for the induced system, Birkhoff’sergodic theorem yields, with convergence in L1pµq,

limNÑ8

1

NSτN pxqφpxq “ lim

NÑ8

1

NSN φpxq “ µprasq´1µpφq .

The claim follows.

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Step 8. Conclusion: any weak Gibbs measure is an equilibrium measure and cpmq “ PGpφq.

Steps 4 and 7 prove that hmpσq `ş

φdm is well-defined and equal to cpmq, which by Step4 is equal to PGpφq, which is equal to suptP pφ, νq : ν P PpΣq and

ş

φ´ dν ă `8u by TheoremA.2, so that m is an equilibrium measure. This completes the proof of Theorem A.4. l

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296 19/12/2016

List of Symbols

8 standard point at infinity r1 : 0s of a projective plane 2281A characteristic function of a subset A 25„ “ „D equivalence relation on index set of an equivariant family D 109||f ||α α-Hölder norm of f P C α

c pZq 42||ψ||` Sobolev W `,2-norm of ψ P C `

c pNq 122| ¨ |v (normalised) absolute value associated to a valuation v 223˚v base point ˚v “ rOv ˆ Ovs of the Bruhat-Tits tree Xv 227

ApOvq maximal compact-open diagonal subgroup of PGL2pKvq 229AutpX, λq automorphism group (edge-preserving, without inversion) of a metric

tree pX, λq36

AutX automorphism group (without inversion) of a simplicial tree X 36

Bpx, rq closed ball of center x and radius r in a metric space 25B˘pw, η1q Hamenstädt’s ball of radius η1 ą 0 with center any geodesic line

extension of w P G˘X31

C geometrically connected smooth projective curve over Fq 223cA complementary set of a subset A 25cpgq period for a system of conductances c of a closed orbit g for the

geodesic flow215

CcpZq space of real-valued continuous maps with compact support on Z 25C α

b pZq space of bounded α-Hölder-continuous real-valued functions on Z 42C k, α

b pZq space of real-valued functions on Z with bounded α-Hölder-continuousderivatives of order at most k along the flow

131

C αc pZq space of α-Hölder-continuous real-valued functions with compact

support on Z42

C k, αc pZq space of real-valued functions on Z with bounded α-Hölder-continuous

derivatives of order at most k along the flow and compact support132

C `c pNq space of real-valued C `-smooth functions with compact support on a

smooth manifold N122

codegDpxq codegree of a vertex x with respect to a subtree D 116codegDpxq codegree of a vertex x with respect to a family of subtrees D 117

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covm,npφ, ψq n-th correlation coefficient of φ, ψ for the measure m under atransformation

122

covµ, tpψ,ψ1q correlation coefficient of ψ,ψ1 under a flow at time t for the measure µ 132

covµv , g correlation coefficient for g P Gv and a measure µv on ΓzGv 242C ΛΓ convex hull in X of the limit set ΛΓ of Γ 26x conjugate of a quaternion x 267ra, b, c, ds crossratio of pairwise distinct points a, b, c, d 273|a, b, c, d|v absolute crossratio of pairwise distinct points a, b, c, d for a valuation v 273

B8X boundary at infinity of X 26BeX set of points at infinity of the geodesic rays whose initial (oriented) edge

is e37

BV D boundary of set of vertices of a simplicial subtree D 116BD maximal subgraph with set of vertices BV D 116B1´D inner unit normal bundle of a closed convex subset D 32B1`D outer unit normal bundle of a closed convex subset D 32

deg v degree of valuation v, equal to dimFq kv 223δ “ δΓ, F critical exponent of pΓ, F q 45∆c˘ Laplacian operator associated to a system of conductances c˘ 92∆x unit Dirac mass at a point x 25DiscD reduced discriminant of a quaternion algebra D over Q 268dD distance-like map on B8X ´ B8D associated with closed convex subset

D238

d “ dp

GXdistance on space of generalised geodesic lines 28

d “ dT 1X distance on space of germs of geodesic lines 29dH Hamenstädt’s distance at infinity associated to an horoball H 31dx visual distance on B8X seen from x P X 27dW˘pwq Hamenstädt’s distance on the strong stable/unstable ball of w P G˘X 30

EX set of edges of a graph X 36ϕRv Euler function of fonction ring Rv 255

f´ negative part of a real-valued map f 25f˘D fibration over B1

˘D with fibers the stable/unstable leaves 33rF potential on T 1X 43F potential on ΓzT 1X 43Fq finite field of order a prime power q 223

g genus of the smooth projective curve C 223p

GX space of generalised geodesic lines in X 28p

GX space of generalised discrete geodesic lines in a simplicial tree X 36p

Geven X space of generalised discrete geodesic lines ` in X with dp`p0q, x0q even 65

298 19/12/2016

GevenX space of discrete geodesic lines ` in X with `p0q at even distance from x0 65G˘X space of generalised positive/negative geodesic rays in X 29G˘, 0X space of generalised geodesic lines in X isometric exactly on ˘r0,`8r 29pgtqtPR (continuous time) geodesic flow on space of generalised geodesics

p

GX 28pgtqtPZ (discrete time) geodesic flow on space of generalised geodesics

p

GX 37

hmpT q metric entropy of a transformation T with respect to a probabilitymeasure m

85

hmpφ1q metric entropy of a flow pφtqtPR with respect to a probability measure m 87hpαq complexity of loxodromic fixed point α 258hpαq complexity of quadratic irrational α in Kv 262hpαq complexity of quadratic irrational α in Qp 270hαpβq relative height of loxodromic fixed point β with respect to α 273hαpβq relative height of quadratic irrational β with respect to α 279HaarKv normalised Haar measure of pKv,`q 225H rts horoball contained in H whose boundary is at distance t from the

boundary of H28

HB`pwq stable horosphere of w P G`X 31HB´pwq unstable horosphere of w P G´X 31H`pwq stable horosphere of w P G`X 31H´pwq unstable horosphere of w P G´X 31

ι antipodal map w ÞÑ tt ÞÑ wp´tqu ofp

GX 28ιGpβq G-reciprocity index of a quadratic irrational β 264IsompXq isometry group of X 26Iv set of classes of fractional ideals of Rv 231

K global function field over Fq 223Kv completion of function field K for the valuation v 223kv residual field of the valuation v on K 223

λpγq translation length of an isometry γ of X 26ΛΓ limit set of a discrete group of isometries Γ of X 26ΛcΓ conical limit set of a discrete group of isometries Γ of X 26L2pY, G˚q Hilbert space of square integrable maps on V Y for the measure volpY,G˚q 38lk links of vertices in simplicial trees 227LΓ length spectrum of action of Γ on X 65ln natural logarithm (with lnpeq “ 1) 25`˘ positive/negative endpoint of geodesic line ` 28`˚ standard basepoint in space of geodesic lines GXv 229

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mDpxq multiplicity of a vertex x with respect to an equivariant family D 116rmF Gibbs measure on the space of geodesic lines GX 55mF Gibbs measure on the quotient space of geodesic lines ΓzGX 55rmF Gibbs measure on the space of discrete geodesic lines GX 62mF Gibbs measure on the quotient space of discrete geodesic lines ΓzGX 62mF renormalised Gibbs measure mF ||mF || on ΓzGX 121mc renormalised Gibbs measure mc||mc|| on ΓzGX 123pµ˘x qxPX (normalised) Patterson density for the pair pΓ, rF˘q 53µW˘pwq skinning measures on the strong stable or strong unstable leaf W˘pwq 106pµHausx qxPX Hausdorff measures of the visual distances dx on ΛΓ 64

NpIq (absolute) norm of a nonzero ideal I in a Dedekind ring 225N˘w homeomorphism between stable/unstable leaves and inner/outer normal

bundles of horoballs32

NεA closed ε-neighbourhood of a subset A of a metric space 25npβq relative norm of quadratic irrational β 261Npxq reduced norm of a quaternion x 268ν¯w conditional measure on the (weak) stable/unstable leaf W 0˘pwq of

w P G˘X107

OxA shadow of a subset A of X seen from x P X Y B8X 26Ov valuation ring of of v in Kv 223

π footpoint projection w ÞÑ wp0q on (generalised) geodesic lines 28πv uniformizer of a valuation v of a function field K over Fq 223πpY, G˚q fundamental groupoid of a graph of groups pY, G˚q 196PD closest point map to a convex subset D 32P˘D closest point map homeomorphism from B8X ´ B8D to outer/inner

normal bundle of D32

rφ˘µ total mass function of Patterson density pµ˘x qxPX 61Pφ pressure of a potential φ under a transformation 85Pψ pressure of a potential ψ under a flow 9, 87Pφpmq metric pressure for a potential φ of a probability measure m invariant

under a transformation85

Pψpmq metric pressure for a potential ψ of a flow-invariant probability measurem

9, 87

Q “ QΓ, F, x, y Poincaré series of pΓ, F q 45qv order of residual field kv 223

Rv affine algebra of the affine curve C´ tvu 224

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βσ Galois conjugate of a quadratic irrational β in Kv 261ασ Galois conjugate of a quadratic irrational α in Qp 270rσ˘D skinning measure on outer/inner normal bundle of convex subset D 103σ´D inner skinning measure on Γz

p

GX of a family of closed convex subsets D 110σ`D outer skinning measure on Γz

p

GX of a family of closed convex subsets D 110rσ˘D outer/inner skinning measure on

p

GX of a family of closed convexsubsets D

109

rσ˘Ω outer/inner skinning measure of a family Ω “ pΩiqiPI of subsets ofpB1˘DiqiPI

109

Tπ first edge map of a geodesic line 37T 1X space of germs at t “ 0 of geodesic lines in X 29trpβq relative trace of quadratic irrational β 261Trpxq reduced trace of a quaternion x 268TvolpY, G˚q volume form on the set of edges of a graph of finite groups pY, G˚q 38TvolpY, G˚, λq volume form of the set of edges of a metric graph of finite groups

pY, G˚, λq38

TVolpY, G˚q total volume of the set of edges of a graph of finite groups pY, G˚q 38TVolpY, G˚, λq total volume of the set of edges of a metric graph of finite groups

pY, G˚, λq39

U ˘D domain of the fibration f˘D 32

V ˘w, η, η1 dynamical neighbourhoods of a point w P G˘X 33V ˘η, η1pΩ

¯q dynamical neighbourhood of a subset Ω¯ of G˘X 33v` germ at t “ 0 of geodesic line ` 29v8 valuation at infinity of FqpY q 222volpY, G˚q volume form on the set of vertices of a graph of finite groups pY, G˚q 38VolpY, G˚q volume of a graph of finite groups pY, G˚q 38V X set of vertices of a graph X 36

w˘ positive/negative endpoint of generalised geodesic line w 28W`pwq strong stable leaf of w P G`X 30W 0`pwq stable leaf of w P G`X 31W´pwq strong unstable leaf of w P G´X 30W 0´pwq unstable leaf of w P G´X 31

|X|λ geometric realization of a metric tree pX, λq 36Xv Bruhat-Tits tree of pPGL2,Kvq 227

ζKpsq Dedekind’s zeta function of a function field K 225

301 19/12/2016

302 19/12/2016

Index

absolutecrossratio, 273value, 222

acylindrical, 76, 149, 212adjacency matrix, 98admissible, 67alc-norm, 240, 243algebraic lattice, 40algebraically locally constant, 240, 243almost precisely invariant, 117anti-reversible, 50antipodal map, 28, 30

bipartite, 36biregular, 36boundary, 116bounded parabolic limit point, 26Bowen ball, 55Bowen-Margulis measure, 55Bowen-Walters distance, 79Busemann cocycle, 27

closest point map, 32cocycle

Busemann, 27Gibbs, 47

codegree, 116, 117cohomologous, 45, 50, 87common perpendicular, 34, 197

ending transversally, 197endpoint, 197multiplicity, 190origin, 197starting transversally, 197

complexity, 258, 262, 270conductance, 50, 197

reversible, 50conical limit

point, 26set, 26

conjugate, 267continued fraction, 266

eventually periodic, 266convergence

narrow, 160weak, 160

convergence type, 45correlation coefficient, 122, 132counting function, 182, 197, 248, 282critical exponent, 45, 52cross-section, 80crossratio, 273

absolute, 273cuspidal ray, 39cylinder, 62, 126

decay of correlationsexponential, 121polynomial, 121superpolynomial, 132

degree, 36Diophantine, 152-Diophantine, 1324-Diophantine, 132discriminant, 268

reduced, 268distance

Bowen-Walters, 79Hamenstädt’s, 30, 31signed, 179visual, 27

distance-like map, 238divergence type, 45doubling measure, 54

uniformly, 54dynamical

303 19/12/2016

ball, 55neighbourhood, 33of a point, 33

edge length map, 36edge-indexed graph, 38ε lc-norm, 43ending transversally, 197endpoint, 197

negative, 28positive, 28

equilibrium measure, 288equilibrium state, 9, 85, 87equivariant family, 108

locally finite, 109Euler function, 255exponential decay

Hölder, 121, 122Sobolev, 122

exponentially mixingHölder, 121, 122Sobolev, 122

extendible geodesics, 26extension, 29

first returnmap, 80time, 80

footpoint projection, 28, 30full, 68function field, 223fundamental groupoid, 196

Galois conjugate, 261generalised geodesic

line, 28discrete, 36

ray, 29segment, 29

geodesiccurrent, 55flow, 28discrete time, 37

line, 28generalised, 28generalised discrete, 36

path, 196geodesically complete, 25

geometric realisation, 36geometrically finite, 26Gibbs

cocycle, 47constant, 68, 288measure, 55, 62weak, 85, 288

property, 56, 62, 68graph

bipartite, 36edge-indexed, 38of groups, 37metric, 38quotient, 38

Greenfunction, 97kernel, 97

growthlinear, 41subexponential, 42

Gurevič pressure, 288

Hamenstädt’sdistance, 30, 31measure, 105

harmonic measure, 97heigth, 230highest point, 230Hölder-continuity, 41

local, 41Hölder norm, 42homogeneous, 149homography, 228Hopf parametrisation, 30

discrete, 37Hopf-Tsuji-Sullivan-Roblin theorem, 56horoball, 28

stable, 31unstable, 31

horosphere, 28stable, 31unstable, 31

index, 212induced

partition, 291system, 291

inner unit normal bundle, 32

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inversion, 36isomorphism, 80iterated partition, 291

Kac formula, 89

Laplacian, 92lattice, 39

algebraic, 40uniform, 39

Ov-lattice, 227leaf

stable, 31strong stable, 30strong unstable, 30unstable, 31

lengthof common perpendicular, 34spectrum, 58, 65

limit pointbounded parabolic, 26conical, 26

linear growth, 41link, 227Lipschitz, 41local field

non-Archimedean, 223locally

constant, 42finite, 109Hölder-continuous, 41Lipschitz, 41

loxodromic, 26, 257reciprocal, 264

Markovchain, 76, 96shiftone-sided, 126transitive, 67two-sided, 67

Markov-good, 76measure

Bowen-Margulis, 55doubling, 54Gibbs, 55Hamenstädt’s, 105Patterson, 53

satisfying the Gibbs property, 56, 62, 68skinning, 103, 106Tamagawa, 269

measured metric space, 54metric

graph of groups, 38pressure, 9, 85, 87tree, 36

modulargraph, 230graph of groups, 230group, 230ray, 231

multiplicity, 116, 181, 190, 197

Nagao lattice, 231narrow topology, 160natural extension, 123

suspended, 134negative

endpoint, 28part, 25

non-Archimedean local field, 223non-backtracking, 149norm, 225

ε lc, 43alc, 240, 243form, 281Hölder, 42of a quaternion, 268of quadratic irrational, 261

order, 268origin, 149, 197ortholength spectrum, 11

marked, 11outer unit normal bundle, 32

Patterson density, 53period, 43, 215, 266Poincaré

map, 80series, 45

pointingaway, 113towards, 113

polynomially mixing, 121positive endpoint, 28

305 19/12/2016

potential, 43, 87associated with, 51cohomologous, 45reversible, 45

precisely invariant, 116almost, 117

pressure, 9, 62, 85, 87, 288Gurevič, 288metric, 9, 85, 87

primitive, 26proper nonempty properly immersed closed

locally convex subset, 182property HC, 8, 44

quadratic irrational, 261complexity, 262reciprocal, 264

radius-continuous ball masses, 167radius-Hölder-continuous ball masses, 167random walk, 97

non-backtracking, 149rapid mixing, 132ray

cuspidal, 39generalised geodesic, 29

reciprocal, 264reciprocity index, 264recurrent, 97reduced, 196regular, 36relative height, 273, 279, 280residual field, 222reversible, 45roof function, 79

shadow, 26lemma, 54for trees, 61

shift, 62, 67, 69, 126signed distance, 179simple, 117simplicial tree, 36skinning measure, 103, 106, 212

inner, 110outer, 110

special flow, 79spherically symmetric, 111

splitting over, 268stable

horoball, 31horosphere, 31leaf, 31

standard base point, 227starting transversally, 197state space, 96strong

stable leaf, 30unstable leaf, 30

subexponential growth, 42subgraph of subgroups, 37subshift of finite type, 67subtree, 26suspension, 79system of conductances, 8, 50

anti-reversible, 50cohomologous, 50reversible, 50

Tamagawa measure, 269terminal vertex, 36Theorem

of Hopf-Tsuji-Sullivan-Roblin, 56thermodynamic formalism, 9topological Markov shift

one-sided, 126transitive, 67two-sided, 67

topologicallymixing, 58, 65transitive, 67

topologynarrow, 160weak, 160

traceof a quaternion, 268of quadratic irrational, 261

transient, 97transition kernel, 96transitive, 67translation

axis, 26, 257length, 26

treebiregular, 36

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cylinder, 62metric, 36uniform, 36

regular, 36simplicial, 36

R-tree, 26

uniform tree, 36uniformiser, 222uniformly doubling, 54unit

normal bundleinner, 32outer, 32

tangent bundle, 29unstable

horoball, 31horosphere, 31leaf, 31

valuation, 221at infinity, 222ring, 221

n-variation, 85n-th vertex of a random walk, 149virtual centre, 211visual distance, 27volume

formof a graph of groups, 38of a graph of groups, 38of a metric graph of groups, 38

of a graph of groups, 38

weak Gibbs measure, 85weak topology, 160

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308 19/12/2016

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Laboratoire de mathématique d’Orsay, UMR 8628 Université Paris-Sud et CNRSUniversité Paris-Saclay, 91405 ORSAY Cedex, FRANCEe-mail: anne.broise@math.u-psud.fr

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Laboratoire de mathématique d’Orsay, UMR 8628 Université Paris-Sud et CNRSUniversité Paris-Saclay, 91405 ORSAY Cedex, FRANCEe-mail: frederic.paulin@math.u-psud.fr

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